Bus 221 Notes
Legend
1. Grey: Particularly important information.
2. Yellow: particularly important procedures
3. Blue: refers to graphics (tables, charts, transparencies of pages, etc.) from the textbook.
4. Black: supplemental material – not covered in the exams or text
Chapter 1: Picturing Distributions with Graphs
1. Individuals and Variables
a. Let’s collect some data from some members of this class.
b. Individuals: “The objects described by a set of data.”
c. Variables: “Any characteristic of an individual.” For example, the height
(variable 1) and weight (variable 2) of the students in our class (the individuals).
i. Quantitative variables: “takes numerical values … [and] recorded in a
unit of measurement …”
ii. Categorical variables: individuals are placed “into one of several groups
or categories.”
d. Sample: a subset of a population, where the population is the full group of
interest.
i. Often data set construction begins with a question that needs to be
answered: What is the mean GPA of a CWU student?
1. The population is defined by the question.
ii. Why might we prefer gathering information from a sample rather than the
full population.
iii. Is there ever a problem with gathering data from a sample?
1. The importance of sample size.
e. Observation: the value that a variable takes for a particular individual.
f. Data: any collection of observations.
g. Example: Transparency: p. 6
i. Individuals: make & model
ii. Variables: vehicle type, transmission type, number of cylinders, city mpg,
highway mpg
2. Categorical Variables: Pie Charts and Bar Graphs
a. Categorical variable: individuals are placed “into one of several groups or
categories” and hence the value of the variable represents a certain category.
b. Distribution of a variable: It “tells us what values [a variable] takes and how
often it takes these values.” That is, it tells us how the data are spread (i.e.,
distributed) across different ranges.
c. Distribution of a categorical variable: It “lists the categories and gives either
the count or the percent of individuals who fall in each category.”
i. Frequency: the number of observations in a range
ii. Percent: the proportion of observations in a range x 100
1. Proportion: number of observations in range / total number of
observations
2. Roundoff error: when the sum of the percents across the different
categories don’t equal 100% because of rounding.
iii. Slide: Example 1.2
1. What are the individuals?
2. What is the variable?
d. Pie chart: the size of the wedge represents the percent of individuals that fit into
a certain category in the sample.
i. Slide: Figure 1.2, p. 8
e. Bar chart: the height of each bar represents the percent or count (number) of
individuals that fit into a given category in the sample.
i. Slide: Figure 1.3, p. 8
ii. The bars are drawn with a space between them.
iii. The height of each bar can represent percent or count.
3. Quantitative Variables: Histograms
a. Quantitative variable: a variable where the value of the variable represents a
quantitative level, rather than a category.
b. Histogram: Very similar to a bar chart. Data are placed into one of a number of
equal sized classes (similar to categories), and then the height of each bar
represents the percent or frequency (number) of individuals that fit into each
class.
i. Similar to a bar chart, except:
1. Rather than categories, there are ranges for the variable.
2. The bars in a histogram do not have any space between them.
ii. Example
1. The raw data: Slide: Table 1.1, p. 12
a. What are the individuals?
b. What is the variable?
2. One histogram: Slide: Figure 1.5, p. 13
3. Another histogram: Slide: Figure 1.6, p. 16
a. “Histograms with more classes show more detail but may
have a less clear pattern.”
4. Interpreting histograms:
a. Overall pattern:
i. Shape
1. Skewed: When “the histogram extends much farther out” in one
direction than another.
a. The distribution can be right skewed or left skewed.
b. Slide: Figure 1.6, p. 16
2. Symmetric: When “the right and left sides of the histogram are
approximately mirror images of each other.”
a. Slide: Figure 1.7, p. 17
ii. Center: For now, “the value with roughly half the observations taking
smaller values and half taking larger values.”
iii. Spread: “For now, we will describe the spread of a distribution by giving
the smallest and largest values.”
b. Deviations from the overall pattern:
i. Outlier: “An individual that falls outside the overall pattern.”
1. Slide: Figure 1.9, p. 19
5. Quantitative Variables: Stemplots:
a. Stemplot: very similar to a histogram (it looks like a sideways histogram), but
which reveals the exact numerical value of sample data in each range.
i. The right most digit is the leaf, and the remaining digits are the stem.
b. Slide: Figure 1.10, p. 20
i. A distribution of the percent of foreign born residents in states using data
from Table 1.1.
c. When making stemplots we often first round the data to the nearest one, ten,
hundered, or thousand. We then put the rounded data into the stemplot.
i. Slide: Example 1.9, p. 21 - 22
d. We will also often split stems, which means divide each stem into two.
i. Slide: Figure 1.11 (and figure under Figure 1.11)
6. Summary – Graphically Representing Distributions
a. Categorical data  Bar Chart or Pie Chart
b. Quantitative data  Histogram or Stem Plot
7. Time Plots
a. Time series data: Sometimes data (on a certain variable) for an individual or
group is collected over time. These type of date are called time series data.
i. Cross sectional data: The name for the data that we have been discussing
up to this point.
1. Cross sectional data on a certain variable is data collected across a
section of individuals at a certain point in time.
2. Bar graphs, pie charts, and histograms plot cross-sectional data.
b. Time plots: a time plot of a variable graphs the level of the variable (on the y
axis) against the time period when the variable was collected (on the x axis).
i. Slide: Figure 1.12, p. 24
ii. Cycles: regular or cyclical up and down movements in data over time.
iii. Trends: a long term movement in one direction over time.
8. Using Statistical Aplets at the Course Website
a. The One Variable Statistical Calculator
Chapter 2: Describing Distributions with Numbers
1. Measuring Center: Mean
a. Mean: Xbar = 1/n ∑xi
i. Xbar: the mean of the variable (the mean of the observations on the
variable in question for the different individuals in our sample) at which
we are looking, where x represents the variable at which we are looking.
ii. n: the number of individuals in the data.
iii. ∑: the sum
iv. x: the variable at which we are looking
v. xi: the observation on the variable at which we are looking, for individual
i.
1. The “observations for the individuals” otherwise known simply as
the “observations”.
vi. ∑xi: the total sum of the observations on the variable for each individual
(individual 1 through individual n).
2.
3.
4.
5.
6.
1. ∑xi ≡ x1 + x2 + … + xn
vii. Notation allows us represent formulas easily.
viii. We can calculate the mean of a sample, or a full population.
Measuring Center: Median
a. Median (M): the midpoint of a distribution
i. “The number such that half the observations are smaller and the other half
are larger”
ii. Finding the Median
1. “Arrange all observations in order of size, from smallest to
largest.”
2. If the number of observations n is odd, the median M is the center
observation in the ordered list. If the number of observations n is
even, the median M is the [mean of]… the two center observations
in the ordered list?”
3. “You can always locate the median in the ordered list of
observations by counting up (n + 1)/2 observations from the start
of the list.”
iii. Slide: Example 2.3, p. 42
1. This stemplot allow us to find the middle number/s.
Comparing the Mean and Median
a. In a skewed distribution, the mean is usually further out in the tail of the
distribution (the side of the distribution that extends further.
Measuring Spread: The Quartiles, five-number summary, & boxplots
a. Quartiles
i. First quartile (Q1): the median of the ordered observations to the left of
the median.
ii. Third quartile (Q3): the median of the ordered observations to the right of
the median.
b. Five number summary: Minimum, Q1, M, Q3, Maximum
c. Boxplot:
i. Slide: Figure 2.1, p. 46
Spotting suspected outliers:
a. Interquartile range (IQR): Q3 – Q1
b. Interquartile Rule for outliers:
i. An observation is a suspected outlier if the observation falls 1.5 x IQR
above third quartile or below first quartile.
Measuring Spread: Standard Deviation
a. Mean absolute deviation
b. s (standard deviation) = sqrt [ 1/(n-1) ∑(xi – Xbar)2 ]
i. Here xi represents the value of the variable for the i’th individual.
c. Slide: Figure 2.2, p. 51
d. s2 (variance) = 1/(n-1) ∑(xi – Xbar)2
e. Calculating s by hand.
i. Make a column of observations
ii. Make a column of deviations: (xi – Xbar)
iii. Make a column of squared deviations: (xi – Xbar)2
iv. Sum the last column and divide by (n – 1) to calculate s2
v. Take the square root of s2 to calculate s.
vi. Example 2.7, p. 50
7. Choosing Measures of Center & Spread
a. The five-number summary is preferred with a skewed distribution or with strong
outliers while the Xbar and s are convenient for somewhat symmetric
distributions.
Chapter 3: The Normal Distribution
1. Density curves:
a. A density curve is another way of representing a distribution for a quantitative
variable. A density curve is just a line (a continuous function).
i. See Figure 3.1, p. 70
ii. For a density curve, the y axis represents proportion, which is just “percent
/ 100.”
iii. For a density curve, the x axis represents different possible values for the
variable at which we are looking.
1. For a histogram, the x axis also represents different possible values
(or ranges) for the variable at which we are looking.
iv. Proportion of data falling between two values is now represented by the
area under the curve (and above the x axis).
1. For a histogram, the proportion of data falling between to values is
represented by the height of the bar.
v. The total area under the curve is 1.
1. For a histogram, the summed heights of the bars is 1.
vi. The proportion for any single value is 0, because the area between any
single value and itself is just 0.
b. Note, contrary to all of the histograms we have been looking at which show the
distribution of a sample, a density curve will typically be used to estimate the
distribution of a full population.
c. Questions
i. How do we calculate the proportion of observations below or above a
certain level?
1. How would you do it for a histogram?
a. See Figures 3.2a, p. 71, for how to do it with a histogram
2. How would you do it for a normal distribution?
a. See Figure 3.2b, p. 71, for how to do it with a density
curve.
ii. How do we calculate the proportion of observations between two values?
1. How would you do it for a histogram.
2. How would you do it for a normal distribution.
iii. What is the height of a horizontal density curve over the region (0, 1)?
How about over the region (0, 2)?
1. See Figure 3.4, p. 73
2. Describing density curves
a. Median: the equal areas point.
i. See Figure 3.5a and 3.5b, p. 73
b. Mean: “the mean is pulled away from the median toward the long tail.”\
i. See Figure 3.5a and 3.5b, p. 73
3. Normal distributions (Normal density curves)
a. A Normal distribution is a symmetric bell shaped density curve described by the
following equation.
i. f (X) 
1
 X  
1 / 2 

  
2
for    X   .
e
 2
ii. You do not need to know this equation for the exam.
iii. This is the equation for a whole family of normal density curves in the
same way that f(X) = mx + b is an equation for a whole family of straight
lines.
1. This might look more familiar as y = mx + b.
iv. I will use the terms “density curve” and “distribution” interchangeably.
b. There is a whole family of Normal density curves. In fact, there are infinitely
many normal density curves. By altering μ and σ the density curve can be made
to take a wide variety of shapes. However, it will always be symmetric and bell
shaped.
c. It turns out that μ is the mean (the center) of the distribution, and σ is the standard
deviation of the distribution.
d. N(μ, σ) is notation for a Normal distribution with mean μ and standard deviation
σ.
i. For example, N(0, 1) would refer to a normal distribution with a mean of 0
and a standard deviation of 1.
ii. μ, the mean, determines the location of the distribution on the x axis.
1. Note that the mean and median are located at the center of a
Normal distribution (at the same point).
a. See Figure 3.5a, p. 73
2. Later we will use μ to describe the mean of the population, which
is the full set of data from which a sample is taken. Xbar is the
mean of a sample.
iii. σ, the standard deviation, determines the spread of the distribution.
1. See Figure 3.8, p. 75
2. The change of curvature points “are located at distance σ on either
side of the mean (μ).
a. See Figure 3.8, p. 75
3. Does it make sense that a more spread Normal density curve would
have a greater standard deviation σ? Remember, the analogy is a
histogram with a greater spread.
4. Later we will use σ to describe the standard deviation of the
population, which is the full set of data from which a sample is
taken. s is the standard deviation of the sample.
e. Note, contrary to all of the histograms we have been looking at which show the
distribution of a sample, the Normal distribution will be used to estimate the
distribution of a full population.
4. Motivation for using the Normal distribution
a. First: “Normal distributions are good descriptions for some distributions of real
data.”
i. However, it is not perfect. It implies that there are some values for x
which are extremely high and extremely low.
b. Second: “Normal distributions are good approximations to the results of many
kinds of chance outcomes, such as the proportion of heads in many tosses of a
coin.”
c. Third: “We will see that many statistical inference procedures based on Normal
distributions work well for other roughly symmetric distributions.”
5. The 68-95-99.7 rule
a. “In [a] Normal distribution with mean μ and standard deviation σ :
i. Approximately 68% of the observations fall within σ of the mean μ.
ii. Approximately 95% of the observations fall within 2σ of μ.
iii. Approximately 99.7% of the observations fall within 3σ of μ.”
iv. This analogizes to a histogram.
v. Figure 3.9, p. 77
vi. Figure 3.10, p. 78
vii. Example 3.3, p. 79
6. Cumulative proportion
a. The “cumulative proportion for” a given variable is the “proportion of individuals
whose observed value is equal to or less than” some specified value for the
variable in question.
i. P(x < “specified value”)
ii. Where P is the proportion of data less than the “specified value.”
b. The cumulative proportion is just the area below the density curve to the left of
the “specified value.”
i. Figure, top of page 82
c. The proportion of individuals whose observed value is greater than x for the
variable in question is just “1 – the cumulative proportion of x.”
i. Example 3.5, p. 82
7. The standard Normal distribution
a. The standard normal distribution is a normal distribution with the below
properties. That is, it is a distribution for a variable that has the following
characteristics:
i. μ = 0
ii. σ = 1
b. Figure 3.9, p. 77
c. Note that the value of any observation tells us exactly how many standard
deviations that observation is from the mean.
i. For example, imagine the observation 2.5 in the standard normal
distribution. Since μ = 0 and σ = 1 it is implied that the value 2.5 in the
standard normal distribution is 2.5 standard deviations from the mean.
d. Conveniently, there is standard normal table in the back of our book on p. 690691 that tells us the cumulative proportion for any observation in a data set that
has a standard normal distribution.
e. Unfortunately, this table only applies to a standard normal distribution, which is
just one of the infinitely many normal distributions.
8. Using the standard Normal table to find proportions for any Normal distribution.
a. It turns out that the cumulative proportion for a specific point (observation) in any
and every Normal distribution is determined by the number of standard deviations
that point is from the mean.
b. Therefore, to find the cumulative proportion of a point (x value) in any and every
Normal distribution, we can simply calculate how many standard deviations the x
value is from the mean, z = (x - μ) / σ, and then look up the number in the standard
normal table on p. 676-677.
i. z = (x – μ) / σ
ii. z is called the “z value” or “z score.” It tells us how far a specific point (x
value, that is, a specific observed value of a variable) falls from the mean
for a Normally distributed variable.
iii. Example 3.7
c. Summary - Finding cumulative proportions for values in any Normal
distribution:
i. Convert the value (observation) to a z value using the formula: z = (x – μ)
/σ
ii. Look up the z value in Table A (p. 676-677) to find the corresponding
proportion.
d. Finding the proportion of data that lie between two particular values. That
is, finding the proportion of data that fall within a certain range, say between x1
and x2.
i. Convert the x values (x1 and x2) to z values.
ii. Look up the z values in Table A and use “the fact that the total area under
the curve is 1 to find the required area under the standard Normal curve.”
iii. Example 3.8, p. 85.
9. A general approach for finding a cumulative proportion for a certain value in a
normal distribution.
a. Draw a picture of a distribution, showing the area for which you are looking.
b. Write out the problem in the following form.
i. P(x < “some value”)
1. When determining the proportion below some value:
ii. 1 – P(x < “some value”)
1. When determining the proportion above some value:
iii. P(x < “some value”) – P(x < “some other value”).
1. When determining the proportion between two values
c. Calculate the z-score (the number of standard deviations away from the mean):
i. Convert an x value from the above step to a z value using the formula:
z = (x – μ) / σ
d. Use the table to look up the corresponding proportion(s):
e. Use the proportions to calculate your answer.
10. A general approach for finding a value for a certain proportion in a normal
distribution.
a. Draw a picture displaying the relevant cumulative proportion (the area to the left
of a certain value).
i. This may be 1 – “the proportion given in the problem” if the problem asks
you to find the value corresponding to a certain proportion above that
value.
b. Find the z-score: Look up the relevant proportion in Table A and find the
corresponding z value.
c. Calculate the x value using the formula.
i. x = μ + z σ
11. Example of the usefulness of the normal distribution
a. If you had sample data on a certain variable for a group of individuals, you could
calculate proportions you were interested in using the following steps:
i. Assume (following a quick confirming glance at the histogram for the sample)
that the variable is distributed Normally.
ii. Estimate the mean and standard deviation of the Normal distribution using the
sample mean and standard deviation.
iii. Use the z formula and the standard Normal table to do such things as calculate
the proportion of individuals in the full population whose value (for the variable
being considered) falls below a certain level.
Chapter 4: Scatterplots and Correlation
1. Explanatory and response variables
a. Explanatory variable (causal or independent variable): a variable which “may
explain or influence changes in the response variable.”
b. Response variable (dependent variable): a variable which “measures an outcome
of a study.” Alternatively, a variable that responds to changes in the explanatory
variable.
c. One way to identify when there is a true explanatory-response relationship is
when a treatment of the explanatory variable causes changes in the response
variable.
d. Example 4.1
e. It is not always obvious whether there is an explanatory-response relationship
between variables.
i. Example: television and health in cross country data
ii. Even when we suspect there is an explanatory-response relationship
between two variables, it is sometimes not obvious which is the
explanatory and which is the response variable.
iii. Example: police and crime in cross city data
f. Correlation, the emphasis of this chapter, measures how two variables are related,
but not whether the relationship is due to an explanatory-response relationship.
2. Displaying relationships between data: scatterplots
a. Up to now we have focused on describing one variable: mean, standard deviation,
histogram (distribution), etc. In the next few chapters we will focus on the
relationship between two different variables.
i. Table 4.1
b. A scatterplot is drawn using observations on two variables from a group of
individuals. Each point on the graph represents the following ordered pair for
each individual: (observation on first variable for a given individual, observation
on second variable for that same individual).
i. Figure 4.2
c. Example – height vs. weight: utilize a table with observations on two variables for
a group of individuals to “fill in” a graph where one variable is represented on the
x axis and the other variable is represented on the y axis.
d. A scatterplot will often give insight into the relationship between two variables.
(That is, a scatterplot will often give insight into how two variables are related).
i. Positive relationship: If a rise in one variable is associated with a rise in
the other.
ii. Negative relationship: if a rise in one variable is associated with a fall in
the other.
iii. No relationship: if a rise in one variable is associated with no particular
change in the other variable.
2. Interpreting scatterplots
a. Overall pattern of relationship
i. Direction of relationship
1. Positive association
2. Negative association
3. Note: if the line is sloped upwards, the relationship is positive. If
the line is sloped downwards, the relationship is negative.
ii. Form of relationship (e.g., linear)
iii. Strength of relationship (strong or weak)
1. The closer the pattern of the data are to a curve / line (that is, the
more compact the formation of the data are giving the appearance
of a curved or straight line), the stronger is the relationship.
b. Deviations from the pattern
i. Outliers
3. Adding categorical variables to scatterplots
a. This is done by using a different plot color or symbol for individuals in each
category.
b. Doing so can provide information useful in understanding data. For example, it
can provide information useful when trying to determine whether the relationship
between variables is causal or not.
i. Graph: % white versus AFDC payments.
c. Figure 4.3.
4. Measuring linear association: Correlation
1 n  xi  x  yi  y 

a. r =

n  1 i 1  s x  s y 
b. Correlation tells us about the direction and strength of the LINEAR relationship
between two variables. (Two variables may be strongly related, but if the
relationship is not close to linear, the correlation will be have a low absolute
value.)
i. r will always be between -1 and +1.
ii. A positive r implies a positive relationship between the variables and a
negative r implies a negative relationship between the variables.
iii. The more closely the pattern of the data seen in a scatterplot resembles a
straight line, the greater the absolute value of r.
c. Figure 4.5.
d. Chapter 4 example (excel spreadsheet at course website)
5. Facts about correlation
a. “Correlation makes no distinction between explanatory and response variables.”
b. “Because r uses the standardized values of the observations, r does not change
when we change the units of measurement of x, y, or both.”
c. “Positive r indicates positive association between the variables, and negative r
indicates negative association.”
d. “The correlation r is always a number between -1 and 1.”
e. “Correlation requires that both variables be quantitative, so that it makes sense to
do the arithmetic indicated by the formula for r.”
f. “Correlation measures the strength of only the linear relationship between two
variables. Correlation does not describe curved relationships between variables,
no matter how strong they are.”
g. “Like the mean and standard deviation, the correlation is not resistant: r is
strongly affected by a few outlying observations.”
h. “Correlation is not a complete summary of two variable data, even when the
relationship between the variables is linear.”
6. When a scatterplot (and a correlation calculation) reveals a relationship between
two variables, it is not necessarily implied that the relationship is causal. Therefore,
a scatterplot (and a correlation calculation) cannot tell us whether the relationship
between the two variables is causal.
a. One variable has a causal effect on another variable when in a properly done
experiment an increase in the causal variable results in a change in the average
level of the other variable.
b. One example of why two variables would be correlated even though there is no
causal relationship is when a third variable omitted from the analysis has a causal
impact on both variables.
c. When making a scatterplot and a causal relationship between the two variables is
suspected, the explanatory variable (causal variable) is placed on the horizontal
axis and the response variable (other variable) is placed on the vertical axis.
i. It is not implied that a variable is explanatory just because it is placed on
the x axis and declared by a person doing a study to be an explanatory
variable. If the “explanatory” variable does not have a causal impact on
the response variable, then it is not an explanatory variable.
d. Examples:
i. Height plotted against weight
1. Which is the explanatory variable and which is the response
variable?
ii. Price of house plotted against price of car
1. Is there “another variable” which is driving this correlation?
iii. Shoe size plotted against number of basketball games played.
1. Is there “another variable” which is driving this correlation?
Chapter 5: Regression
1. Regression Line: “A straight line that describes how a response variable y changes
as an explanatory variable x changes.”
a. Explanatory variable: the x variable.
b. Response variable: the y variable.
c. Figure 5.1.
d. Basically, a regression line is the equation of the line which comes as close as
possible to matching the data in the scatterplot.
e. The explanatory variable does not necessarily cause changes in the response
variable.
i. Example:
1. Explanatory variable: expensive restaurant dinners per month.
2. Response variable: price of car
2. The Least-Squares Regression Line: the least squares regression line of y on x is the
line that makes the sum of the squares of the vertical distances of the data points
from the line as small as possible.
a. Figure 5.5.
b. y-hat = a + bx
i. This is a linear relationship because it has the general form of a line: y-hat
= mX + b. a above corresponds with b here, and bX above corresponds
with mX here.
ii.
iii.
c. a: mathematically, a is the y axis intercept of the line. Conceptually, it is the
value of the response variable y when the explanatory variable x = 0.
d. b: mathematically, b is the slope of the line. Conceptually, it is the amount that
the response variable y changes in response to a 1 unit change in the explanatory
variable x.
e. For any level of x, the height of the line tells us the predicted level of x. y-hat
from the above equation tells us the same thing.
f. We use a linear regression to do the following:
i. b: Estimate the amount that y changes in response to a 1 unit change in x.
ii. y-hat: Predict level of y for any given level of x.
1. To get a prediction of y, just plug x into the least-squares
regression equation: y-hat = a + bx
g. See Chapter 5 Example Problem at course website.
3. Facts about least-squares regression
a. Fact 1: “The distinction between explanatory and response variables is essential in
regression.”
i. If we switch the variables on the x and y axis, we get a different b.
b. Fact 3: r2 is the square of the correlation. It is the fraction of the variation in the y
values that is explained by the regression of y on x.
i. A higher r2 implies that we have a “better” prediction of y.
4.
5.
6.
7.
ii. However, r2 does not tell us anything about how accurate the estimate of
b is, where b is the estimated impact of a one unit change of x on y.
iii. If there were two sets of data that both had the same regression line, the
data set with a scatterplot that had a pattern of data more closely
resembling a line would have the higher r2.
Residuals: “The difference between an observed value of the response variable and
the value predicted by the regression line.”
a. residual = observed y – predicted y
b. residual = y – y-hat
c. Figure 5.2.
d. Figure 5.5.
e. On the graph, this is just the vertical distance between the predicted level of Y for
a given level of X (Y-hat, which is the height of the line which most closely
matches the data, for a given level of X: Y-hat = a + bX), and the actual level of Y
for that same level of X.
f. “The mean of the least squares residuals is always zero.”
i. This is one of the conditions for the least squares minimization.
g. “A residual plot is a scatterplot of the regression residuals against the explanatory
variable. Residual plots help us assess how well a regression line fits the data.”
i. Figure 5.6.
Influential observation: an observation that noticeably changes the level of b when
removed.
a. Note that not all outliers are influential.
i. Figure 5.7.
ii. Figure 5.8.
Cautions about correlation and regression
a. “Correlation and regression lines describe only linear relationships.”
b. “Correlation and least-squares regression lines are not resistant to outliers.”
c. Beware of predictions made from extrapolation
i. Extrapolation: using a regression line to predict the value of a y variable
when the level of the x variable is far outside the range of the actual x
data.
Association Does Not Imply Causation
a. “An association between an explanatory variable x and a response variable y,
even if it is very strong, is not by itself good evidence that changes in x actually
cause changes in y.”
i. That is, “a strong association between two variables is not enough to draw
conclusions about cause and effect.”
ii. “The best way to get good evidence that x causes y is to do an
experiment.”
b. “The relationship between two variables can often be understood only by taking
other variables into account. Lurking variables can make a correlation or
regression misleading.”
i. Lurking variable: “A lurking variable is not among the explanatory or
response variables in a study and yet may influence the interpretation of
relationships among those variables.”
ii. You should always think about possible lurking variables before you draw
conclusions [on causality] based on correlation or regression.”
Chapter 8: Sampling (to get accurate data from which estimates can be made)
1. Introduction
a. Before we can answer any statistical questions and or perform any statistical analysis
(like calculating a regression estimate of b, or calculating the mean for a variable) we
need to collect (or have someone else collect) data. This chapter is about the process by
which we collect data.
2. Population versus sample
a. “The Population in a statistical study is the entire group of individuals about which we
want [to acquire] information.”
b. A Sample is a subset of the population from which we actually collect information.
i.
The sample is typically utilized as an alternative to the population because it is too
difficult or costly to collect data on the entire population.
c. Sampling design: the process by which a sample is chosen from the population.
d. Sample survey: a survey given to a sample that is used to acquire information about a
population.
e. The goal of a sample is for it to be reflective of a population so that it can yield accurate
information about the population. However, many samples because of the way in which
they are chosen are not reflective of the population.
3. Sample Bias – How to sample badly
a. Biased Sample: A sample that systematically favors certain individuals from the
population and hence systematically favors certain outcomes – for example,
systematically favors a higher or lower mean of a variable.
i.
A sample that is drawn from a non-representative subset of the population is often
biased.
b. Convenience sample: “A sample selected by taking the members of the population that
are easiest to reach.”
i.
Convenience samples are often biased samples.
4. Simple random Samples
a. Simple Random Sample (SRS): A sample where each member of the population has an
equal probability of being selected, and every possible sample has an equal probability of
being selected.
i.
An SRS is one type of probability sample.
b. A random sample is unbiased. It does not systematically favor any individuals from the
population. That is, we expect that a random sample is not biased towards (i.e., does not
contain “too much” of) any particular type of individual from the population.
c. Note, however, that even random samples may not be representative of the population.
The reason is that even if a sample is chosen randomly, by chance the sample still may
end up over-representing certain types of individuals from the population, even if we
didn’t expect the sample to over-represent any group. This is particularly true with small
samples.
5. Inference about the population
a. “The purpose of a sample is to give us information about a larger population. The
process of drawing conclusions about a population on the basis of sample data is called
inference because we infer information about the population from what we know about
the sample.”
b. Unfortunately, “it is unlikely that results from a random sample are exactly the same as
for the entire population … the sample results will differ somewhat, just by chance.”
c. “Properly designed samples avoid systematic bias, but their results are rarely exactly
correct and they vary from sample to sample.”
d. “One point is worth making now: larger random samples give more accurate results than
smaller samples.”
6. Other sampling designs
a. Stratified random sample: “First classify the population into groups of similar
individuals, called strata. Then choose a separate SRS in each stratum and combine these
SRSs to form the full sample.”
i.
This is another kind of probability sample.
ii. Stratified random samples are often used to get a more accurate description of small
groups. By “oversampling” a small group, we can get a more accurate description of
that group.
1. Note that before the oversampled group (strata) is combined with the other
stratum, it must be weighted appropriately.
7. Cautions about sample surveys (Potential sources of bias in a sample)
a. Undercoverage: When the chosen sample excludes a group (or groups) from the
population.
b. Nonresponse: “when an individual chosen for the sample can’t be contacted or refuses to
participate.”
c. Response bias: When an individual’s response to questions are biased. Examples of
response bias include the following.
i.
“People know that they should take the trouble to vote, for example, so many who
didn’t vote in the last election will tell an interviewer that they did.”
ii. “The race or sex of the interviewer can influence responses to questions about race
relations or attitudes toward feminism.”
iii. “Answers to questions that ask respondents to recall past events are often inaccurate
because of faulty memory.”
d. Wording of questions: “Confusing or leading questions can introduce strong bias, and
changes in wording can greatly change a survey’s outcome.”
Chapter 9: Experiments (to measure causal effects)
1. Introduction
a. The principles in this chapter pertain only to samples taken to analyze the effect
of one variable on another (for example, the effect of education on earnings).
b. The principles in chapter 8 pertain to both samples taken to analyze certain
variables (for example the mean height of a CWU student) and samples taken to
analyze the effect of one variable on another (for example, the effect of education
on earnings).
2. Experimental studies versus observational studies
a. Explanatory variable: The treatment is referred to as the explanatory variable.
b. Response Variable: The effect being measured is referred to as the response
variable.
3.
4.
5.
6.
c. Experimental study: a group of study participants is divided up into a treatment
group (or groups) and a non-treatment group (control group). The treatment
group is then given a treatment. The effect of the treatment on the response
variable is then compared across the two groups.
i. The experimenter decides who receives the treatment.
d. Observational study: data are collected from members of the population who
have received varying degrees of the treatment variable, but where the treatment
decision was not made by the experimenter. The effect of the treatment
(explanatory variable) on the response variable is then analyzed.
i. The experimenter does not decide who receives the treatment.
Control: The Key Element Behind The Difference Between Experimental and
Observational Studies
a. The Relationship Between Confounding Lurking Variables and Control
Potential problems with observational studies
a. Lurking variable: “A lurking variable is not among the explanatory or response
variables in a study and yet may influence the interpretation of relationships
among those variables.”
b. Confounding: “Two variables (for example, an explanatory variable and a
lurking variable) are confounded when their effects on a response variable cannot
be distinguished from each other.”
c. Observational studies often have confounding lurking variables due to differences
in characteristics between the treatment and control group. When there is a
confounding lurking variable, the measured effect of the explanatory variable on
the response variable is biased.
d. Figure 9.1
Experiments
a. Vocabulary
i. Subjects: the individuals studied in an experiment.
ii. Factors: the explanatory variables in an experiment.
1. Factors can be combined to make up a treatment.
2. Different degrees of a factor (for example, different amounts of a
drug) can be used to create different treatments.
iii. Treatment: “Any specific experimental condition applied to the subjects.
If an experiment has several factors, a treatment is a combination of
specific values for each factor.”
1. Last chapter the treatments were get a factor or don’t get a factor.
This chapter, a treatment can consist of different combinations of
factors/non-factors.
b. Visual representation of combining factors to get different treatments.
i. Example 9.3.
ii. Figure 9.2
c. Advantages
i. Well run experiments often do not have confounding lurking variables.
Therefore, they avoid the bias created by confounding lurking variables.
ii. “We can study the combined effects of several factors simultaneously.”
How to experiment badly
a. Don’t randomly select who receives the treatment (or the different treatments).
When individuals aren’t randomly chosen to receive the treatment, experiments
become susceptible to the bias created by confounding lurking variables.
i. “A simple design often yields worthless results because of confounding
with lurking variables.”
7. The logic of randomization in comparative experiments
a. “Random assignment of subjects forms groups that should be similar in all
respects before the treatments are applied.”
b. “Comparative design ensures that influences other than the experimental
treatments operate equally on all groups.”
c. “Therefore, differences in average response must be due either to the treatments
or to the play of chance in the random assignment of subjects to the treatments.”
i. Statistical significance: “An observed effect so large that it would rarely
occur by chance is called statistically significant.”
ii. “If we assign many subjects to each group … the effects of chance will
average out and there will be little difference in the average responses in
the two groups unless the treatments themselves cause a difference.”
8. How to experiment well: Randomized comparative experiments: “An experiment that
uses both comparison of two or more treatments and chance assignment of subjects to
treatments…”
a. Completely randomized experiment: “All the subjects are allocated at random
among all the treatments”
i. Visual representation of a completely randomized experiment:
1. Figure 9.3
b. Control group: The group in an experiment that receives no treatment, or that
receives an alternative treatment to which the treatment being analyzed is being
compared.
i. Another visual representation of a completely randomized experiment:
1. Figure 9.4
9. Cautions about experimentation
a. “The logic of a randomized comparative experiment depends on our ability to
treat all the subjects identically in every way except for the actual treatments
being compared.”
i. Placebo: a placebo is a dummy, or fake, treatment. Placebos are useful in
experiments because individuals given a treatment often show an effect
just because of the belief that an effect should occur. To control for this
affect, the control group is given a placebo.
ii. Double-blind: an experiment is double blind when neither the individuals
receiving the treatments, nor the scientist analyzing the effects of the
treatment, are aware of who received the actual treatment and who
received the placebo. Double blind experiments are useful because often
scientists recording effects will record an effect if they expect to see an
effect.
iii. Lack of realism: When “the subjects or treatments or setting of an
experiment … [do] not realistically duplicate the conditions we really
want to study.” An unrealistic environment often influences the degree of
the effect of a treatment.
10. Principles of experimental design
a. Randomize
b. Control Group (for comparison purposes)
c. Use enough subjects (i.e., have a large sample size)
d. Other
i. Placebo
ii. Double- Blind
iii. Realism
11. Matched pairs and other block designs
a. Matched pair design: “A matched pairs design compares just two treatments.
Choose pairs of subjects that are as closely matched as possible. Use chance to
decide which subject in a pair gets the first treatment. The other subject in that
pair gets the other treatment. That is, the random assignment of subjects to
treatments is done within each matched pair, not for all subjects at once.
Sometimes each “pair” in a matched pairs design consists of just one subject, who
gets both treatments one after the other. Each subject serves as his or her own
control. The order of the treatments can influence the subject’s response, so we
randomize the order for each subject.”
b. Block design: There is non-random assignment into blocks, or groups (for
example men and women), before there is random assignment of treatments.
i. Figure 9.5
12. Homework
a. Note, any time randomization is required for a problem, please use the Simple
Random Sample applet at the textbook website.
Chapter 10: Introducing Probability
1. The idea of probability
a. Random: “We call a phenomenon random if individual outcomes are uncertain
but there is nonetheless a regular distribution of outcomes in a large number of
repititions.”
i. Alternatively: We call a phenomenon random if individual outcomes are
uncertain but there is a distribution which describes the probability of
different possible outcomes.
ii. Example 10.2: the proportion of heads is random. It is uncertain what the
number of heads will be.
1. Nonetheless there is a distribution over the different possible
proportions for a given number of tosses.
b. Random variable: “A variable whose value is a numerical outcome of a random
phenomenon.”
c. Probability: The likelihood that something will occur.
i. “The probability of any outcome of a random phenomenon is the
proportion of times the outcome would occur in a very long series of
repititions.”
ii. Example 10.2: the probability of getting a “head” is approximately 0.5.
But what is it exactly?
1. Figure 10.1
2. How do you know the probability of getting a head is 0.50? Is it
really 0.50.
3. What is the probability of getting a head in a weighted coin?
4. Note the great degree of variability in the percentage of heads at
smaller sample sizes.
5. Note that the degree of variability in the percentage of heads
diminishes with larger sample sizes.
6. How many repititions (how large a sample size) do you need to
know whether you have an accurate estimate of the probability?
iii. The law of large numbers: the idea that the proportion/percentage
approaches the probability as the number of trials increases.
1. Figure 10.1
d. Digression: The law of large numbers applies to the sample mean:
i. Vegetable ratings: How should we sort?
ii. Finding a lawn and garden gift for your parents at Amazon.com: why does
Amazon filter out products with small sample sizes?
e. Probability distribution: “The probability distribution of a random variable X
tells us what values X can take [think of the different values as different events]
and how to assign probabilities to those values.”
2. A new interpretation of distributions.
a. Probability vs proportion/percentage:
i. Probability and proportion/percentage are closely linked. If the
proportion/percentage of times that the event occurred in the population
was PROB, then the probability of the event occurring in a random sample
would be PROB.
1. For example, if the proportion of CWU students below 68 inches
in height is 0.50, then if we randomly selected one CWU student
there would be a 50 percent chance (0.50 probability) of the
student having a height below 68 inches.
b. In chapter 1, distributions represented the percent of data in a certain range or
category.
c. Now, a distribution represents the probability of getting a certain outcome FOR
ONE OBSERVATION.
d. This new interpretation is the most practical application of distributions for you
personally. It has the most relevance to your everyday decisions. The outcome
associated with virtually every decision that you make is a random variable.
e. Examples:
i. Medical decisions and surgical outcomes.
ii. Purchase decisions and product quality outcomes (in an environment of
poor quality control).
iii. The college decision and your salary outcome?
3. Continuous probability models
a. Definition: A probability model with a continuous sample space is called
continuous.
i. “A continuous probability model assigns probabilities as areas under a
density curve. The area under the curve and above any range of values is
the probability of an outcome in that range.”
ii. Example: the uniform density curve.
1. “The uniform density curve spreads probability evenly between 0
and 1.”
2. Figure 10.5
3. Calculate the probability of being between 0 and 0.5.
iii. Note: “The probability model for a continuous random variable assigns
probabilities to intervals of outcomes rather than to individual outcomes.
In fact, all continuous probability models assign probability 0 to every
individual outcome. Only intervals of values have positive probability.”
iv. Example: The Normal density curve.
1. “Normal distributions are probability models.”
2. Figure 10.6
Chapter 11: Sampling Distributions
1. Parameters and statistics
a. Parameter: “A number that describes the population. In statistical practice,
the value of a parameter is not known because we cannot examine the entire
population.
i. μ
ii. p
b. Statistic: “A number that can be computed from the sample data without
making use of any unknown parameters.”
i. Often times, statistics are used to estimate unknown population
parameters. For example, we use the sample mean to estimate the the
unknown population parameter – the population mean.
ii. Xbar
iii. Phat
c. The emphasis in this class is using the statistic Xbar (the sample mean) to
estimate the parameter μ (the population mean).
2. Statistical estimation and the law of large numbers.
a. “Because [even] good samples are chosen randomly, statistics such as Xbar are
random variables.”
b. Law of large numbers: “Draw observations at random from any population with
finite mean μ. As the number of observations drawn increases, the mean Xbar of
the observed values gets closer and closer to the mean μ of the population.”
i. Figure 11.1
3. Sampling Distributions – The Intuition
a. The Objective – Estimating the population mean: The ultimate objective in this
course is to acquire an estimate of the population mean (the parameter). Will will
use Xbar (a statistic) as an estimate of the population mean. But it is only an
estimate.
4.
5.
6.
7.
b. Xbar is a random variable: we don’t know who will end up in our sample used
to calculate Xbar. Therefore Xbar is a random variable.
c. Xbar has a distribution: as a random variable, Xbar has a distribution. We call
Xbar’s distribution a “sampling distribution.”
d. The center of the distribution of Xbar: Where do you think it is?
e. The spread (standard deviation) of the distribution of Xbar: what do you
think happens to it if the sample size increases?
f. The shape of the distribution of Xbar: How would we go about determining the
shape of the distribution of Xbar.
i. How did we find the shape of the distribution of X in chapter 1.
ii. How should we find the shape of the distribution of Xbar now.
g. The shape of the distribution of Xbar: The Central Limit Theorem (see
below).
Sampling distributions
a. Definition: “The distribution of values taken by the statistic in all possible
samples of the same size from the same population.”
b. Example: Xbar
i. Xbar is a statistic, it is the sample mean.
ii. There is a distribution on Xbar. That is, Xbar can take any one of a
number of possible values, each with a certain probability of occurring.
c. Figure 11.2
The distribution (sampling distribution) of Xbar: shape, center, & spread.
a. “Suppose that Xbar is the mean of an SRS [simple random sample] of size n
drawn from a large population with mean μ and standard deviation σ. Then the
sampling distribution of Xbar has mean μ and standard deviation σ/(square root
n).”
i. Mean of Xbar (μxbar): μ
1. Where μ is the mean of x.
2. “Because the mean of Xbar is equal to μ, we say that the statistic
Xbar is an unbiased estimator of the parameter μ.”
ii. Standard Deviation of Xbar (σxbar): σ / “square root n”
1. Where σ is the standard deviation of x.
2. Figure 11.3
3. The likelihood that the sample mean falls close to the population
mean is determined by the spread of the sampling distribution.
The shape of the distribution of Xbar – 3 cases.
a. Case 1: If x has the N(μ, σ) distribution, then the sample mean Xbar of an SRS of
size n will approximately have the following distribution: N(μ, σ/”square root
n”).
b. Case 2: In reality, for small samples, the shape of the distribution of Xbar is
similar to the distribution of X, whatever that distribution looks like, normal or
not.
c. Case 3: It turns out that the larger the sample, the more the shape of the
distribution looks like a Normal distribution, regardless of the distribution of X.
d. Figure 11.4
The Central Limit Theorem and the shape of the sampling distribution
a. If n is large for an SRS from any population with mean μ and finite standard
deviation σ, Xbar is approximately N(μ, σ/“square root n”), regardless of the
distribution of x.
i. Figure 11.4
ii. Figure 11.5
iii. Figure 11.6
8. Chapter 3 Review
a. It is very useful to review Chapter 3 before doing your homework assignment, as
many of the principles in the chapter are applications of the same principles
learned in Chapter 3.
Chapter 14: Confidence Intervals – The Basics
1. Introduction
a. Statistical inference: “Statistical inference provides methods for drawing
conclusions about a population from sample data.”
i. Statistical inference allows us to “infer” information about the population
from the sample data.
b. Simple Conditions For Inference About A Mean.
i. “We have an SRS from the population of interest. There is no
nonresponse or other practical difficulty.”
ii. “The variable we measure has a perfectly Normal distribution N(μ, σ) in
the population.”
iii. “We don’t know the population mean μ. But we do know the population
standard deviation σ.”
2. The Reasoning of Statistical Estimation
a. The 68-95-99.7 rule for Normal distributions says that in 95% of samples:
i. Xbar will fall between μ + 2 x σxbar and μ – 2 x σxbar
1. Another way of writing this range is:
a. μ ± 2 x σxbar
b. Another way of saying the same thing is that in 95% of samples, μ will fall within
the range:
i. Xbar ± 2 x σxbar
ii. Figure 14.1: sample means fall somewhere in the sampling distribution.
iii. Figure 14.2: approximately 95% of the above defined intervals around the
sample means will capture the unknown mean μ of the population.
3. Margin of Error and Confidence Interval
a. A confidence interval capture the concept that the mean from a sample is a less
accurate estimate (has more variability) when the sample size is smaller.
b. See Figure 10.1.
c. The form of a confidence interval:
i. estimate ± margin of error
1. “A confidence level C, … gives the probability that the interval
will capture the true parameter value in repeated samples.”
2. Xbar ± 2 x σxbar is an example of a 95% confidence interval.
d. Interpreting a confidence interval:
i. “The confidence interval is the success rate of the method that produces
the interval.”
ii. “We got these numbers using a method that gives correct results 95% of
the time.”
4. Confidence Intervals for a Population Mean:
a. Xbar ± z* x (σx / “square root n”)
i. Note: σxbar = σx / “square root n”)
b. z*, called the critical value, is the number of standard deviations that a confidence
interval of size C will fall from the mean.
i. Figure 14.3
c. To find z*, do the following:
i. The easiest way to find z* is to look it up in z* row near the bottom of
Table C.
5. Confidence intervals: the four-step process to solving confidence interval problems
a. State
i. “We are estimating the mean …” (Identify the variable here.)
b. Formulate
i. “We will estimate the mean μ using a confidence interval of size C.” (C is
whatever size confidence interval you are asked to calculate.)
c. Solve
i. Check the conditions for the test you plan to use.
1. SRS
2. Normally distributed random variable
3. Known σ
ii. Calculate the confidence interval.
1. Xbar ± z* x (σx / “square root n”)
a. Find z* from z* row near the bottom of Table C.
d. Conclude
i. “We are C% confident that the mean … is between … and …” (Use the
range implied from the confidence interval calculated above.)
6. How Confidence Intervals Behave
a. Margin of error
i. z* x (σx / “square root n”) is called the margin of error.\
Chapter 15: Tests of Significance (Hypothesis Tests) – The Basics
1. The Reasoning of Tests of Significance (Hypothesis Tests)
a. We make a hypothesis on a population parameter, and then test whether the
sample statistic (e.g., Xbar) is consistent with that hypothesis.
i. Make a hypothesis
ii. Center the hypothesized distribution of Xbar on μ.
iii. Accept or reject the hypothesis based on how far Xbar lies from the center
of the sampling distribution.
2. Stating Hypothesis
a. Start with some finding for which you are trying to find evidence. That is, start
with a claim about the population for which you are trying to find evidence.
b. Formulate an Alternative Hypothesis: Frequently, but not always, “… the claim
about the population that we are trying to find evidence for…”
i. For example, if you are trying to find evidence that something has a non
zero effect, the hypothesis is that there is zero effect.
ii. Denoted Ha
iii. A claim about a population parameter.
iv. An alternative hypothesis always takes one of the following forms:
1. μ ≠ “hypothesized value”
a. This is a two-sided hypothesis.
b. Example: testing whether pizza the night before an exam
has any effect on exam performance.
2. μ > “hypothesized value”
a. This is a one-sided hypothesis
b. Example: testing whether a college education has a positive
effect on earnings.
3. μ < “hypothesized value”
a. This is a one-sided hypothesis
v. “The alternative hypothesis is one-sided if it states that a parameter is
larger than or smaller than the null hypothesis value. It is two-sided if it
states that the parameter is different from the null value.”
c. Formulate a Null Hypothesis: “The statement being tested…”
i. Denoted Ho
ii. It is a claim about a population parameter.
iii. It is the opposite of Ha.
iv. The book always uses the first below form for the null hypothesis,
however, more typically a null hypothesis take one of the following forms.
1. Ho: μ = “hypothesized value”
2. Ho: μ > “hypothesized value”
3. Ho: μ < “hypothesized value”
d. Note: when the alternative hypothesis is the claim about the population for which
you are trying to find evidence, rejecting the null is evidence in favor of your
hypothesis.
3. P values & Statistical Significance
a. P Value
i. Definition: “The probability, computed assuming that Ho is true, that the
test statistic would take a value as extreme or more extreme than that
actually observed is called the P-value of the test. The smaller the Pvalue, the stronger the evidence against Ho provided by the data.”
ii. Alternative Definition: the probability, assuming Ho is true, of getting a
value for Xbar that is any further from Ho.
b. Statistical Significance
i. Definition: “If the P-value is as small or smaller than α, we say that the
data are statistically significant at level α.”
ii. When we say a result is “statistically significant at level α” we are saying
that our rejection of the null hypothesis is significant at level α.
iii. Note that “significant in the statistical sense does not mean important. It
means simply not likely to happen by chance.”
4. Tests for a population mean
a. Test Statistic
i. Definition: a statistic that we construct to test the null hypothesis.
ii. “The test statistic for hypotheses about the mean μ of a Normal
distribution is the standardized version of xbar.” It is known as the z
statistic.
1. z = (Xbar – μo) / (σx / “square root n”)
a. μo, the hypothesized value of μ in the null, is used because
the test statistic is calculated under the assumption that the
null hypothesis is correct.
2. The z statistic tells us how many standard deviations the observed
value of Xbar is from the hypothesized value for μ.
5. Tests of significance: the four step process
a. State
i. State the problem in terms of a specific question about the mean.
1. “We are interested in knowing whether there is evidence that the
mean … is …”
b. Formulate
i. Identify what μx is the mean of:
1. μx is the mean … (Identify the variable here.)
ii. State the alternative hypothesis Ha.
1. Note: Ha will be the claim about the population about which we are
trying to find evidence. It should be a restatement of the “State”
step above, and should take one of the below forms.
2. Ha:
a. μx not = “some number,” or
b. μx > “some number,” or
c. μx < “some number”
iii. State the null hypothesis Ho.
1. Ho:
a. μx = “same number in Ha”
b. μx < “same number in Ha”
c. μx > “same number in Ha”
2. Note: the book always puts μx = “same number in Ha”, but this can
be confusing for students so you should not do it.
c. Solve
i. Check the conditions for the test you plan to use.
1. SRS
2. Normally distributed random variable
3. Known σx
ii. Calculate the test statistic.
1. z = (Xbar – μo) / (σx / “square root n”)
iii. Draw a graph of the sampling distribution assuming that the null
hypothesis is true and identify your value for Xbar in the distribution,.
iv. Find the p value.
1. If Ho is μx = “same number in Ha”
a. and z > 0, then p value = 2 x (1 – “value in Table A”)
b. and z < 0, then p value = 2 x “value in Table A”
2. If Ho is μx > …
a. p value is “value in table A”
3. If Ho is μx < …
a. p value is 1 – “value in table A”
d. Conclude
i. Describe your results in the context of the stated question.
1. If p < α: “We find evidence against the null hypothesis that …”
2. If p > α: “We do not find evidence against the null hypothesis that
…”
3. Note: α is the significant level. When α not given, use 0.05.
Chapter 18: Inference about a population mean
1. Conditions for inference
a. The data are an SRS (simple random sample) from the population.
b. The population distribution for the underlying random variable, x, is
approximately Normal.
i. The larger is the sample size, the less important is the Normality
condition.
ii. “In practice, it is enough that the distribution be symmetric and singlepeaked unless the sample is very small.”
2. Estimating the standard deviation of the sampling distribution
a. When we don’t know the population distribution standard deviation σ, we use s,
the sample standard deviation, as an estimate.
3. The new test statistic: t rather than z
a. t = (Xbar – μo) / (s / “square root n”)
b. t has the t distribution with (n – 1) degrees of freedom.
4. Matched pairs t procedures
a. Definition: “To compare the responses to the two treatments in a matched pairs
design, find the difference between the responses within each pair.” This will
create a new variable. You should use this new variable when doing your test of
significance.
5. Robustness of t procedures
a. “A confidence interval or significance test is called robust if the confidence level
or P-value does not change very much when the conditions for use of the
procedure are violated.”
i. For example, even with outliers, the confidence level and significance test
results will still be robust if the sample size is large enough.
6. Rules of thumb regarding robustness of t procedures
a. “Except in the case of small samples, the condition that the data are an SRS from
the population of interest is more important than the condition that the population
distribution is Normal.
b. Sample size less than 15: Use t procedures if the data appear close to Normal
(roughly symmetric, single peak, no outliers). If the data are skewed or if outliers
are present, do not use t.
c. Sample size at least 15: The t procedures can be used except in the presence of
outliers or strong skewness.
d. Large samples: The t procedures can be used even for clearly skewed
distributions when the sample is large, roughly n > 40.”
7. Tests of significance: the four step process
a. State
i. State the problem in terms of a specific question about the mean.
1. “We are interested in knowing whether there is evidence that the
mean … is …”
2. Note, for matched pair sample designs, the specific question will
be about mean of a new variable you must create from the matched
pair data. The new variable you create will typically be the
difference between two variables for which you have observations.
b. Formulate
i. Identify what μx is the mean of:
1. μx is the mean … (Identify the variable here.)
ii. State the alternative hypothesis Ha.
1. Note: Ha will be the claim about the population for which we are
trying to find evidence. It should be a restatement of the “State”
step above, and should take one of the below forms.
2. Ha:
a. μx not = “some number,” or
b. μx > “some number,” or
c. μx < “some number”
iii. State the null hypothesis Ho.
1. Ho:
a. μx = “same number in Ha”
b. μx < “same number in Ha”
c. μx > “same number in Ha”
2. Note: the book always puts μx = “same number in Ha”, but this can
be confusing for students so you should not do it.
c. Solve
i. Check the conditions for the test you plan to use.
1. SRS
2. Robustness of hypothesis tests using t. (In order to determine
robustness, a histogram or stemplot must often be drawn for
sample sizes less than 40.)
a. For a sample size less than 15: t procedures (like
hypothesis tests) are robust when the data appear close to
Normal (roughly symmetric, single peak, no outliers). If
the data are skewed or if outliers are present, t procedures
are not robust.
b. For a sample size from 15 to 39: t procedures are robust
except in the presence of outliers or strong skewness.
c. For a sample size of 40 or above: t procedures are robust
even for clearly skewed distributions.
ii. Calculate the test statistic.
1. t = (Xbar – μo) / (sx / “square root n”)
iii. Find the p value.
1. Find the p value from Table C. First, you need to find the row that
is closest to the number of “degrees of freedom”, where the
degrees of freedom are n – 1. Then find the two columns which
the absolute value of your t value falls between. The p value will
be between the numbers in one of the last two rows of the table
which correspond with those two columns. Of the two last rows,
choose the one that corresponds with your Ha. If Ha has a > or < in
it, then use the 2nd to last row. Alternatively, if Ha has a “not =” in
it, use the last row.
d. Conclude
i. Describe your results in the context of the stated question.
1. If p < α: “We find evidence against the null hypothesis that …”
2. If p > α: “We do not find evidence against the null hypothesis that
…”
3. Note: α is the significant level. When α not given, use 0.05.
8. The new confidence interval
a. “A level C confidence interval for μ is:”
i. Xbar + t * s / “square root n”
ii. t has the t distribution with (n – 1) degrees of freedom.
9. Confidence intervals: the four-step process to solving confidence interval problems
a. State
i. “We are estimating the mean …” (Identify the variable here.)
b. Formulate
i. “We will estimate the mean μ using a confidence interval of size C.” (C is
whatever size confidence interval you are asked to calculate.)
c. Solve
i. Check the conditions for the test you plan to use.
1. SRS
2. Robustness of confidence intervals using t. (In order to determine
robustness, a histogram or stemplot must often be drawn for
sample sizes less than 40.)
a. For a sample size less than 15: t procedures (like
confidence intervals) are robust when the data appear close
to Normal (roughly symmetric, single peak, no outliers). If
the data are skewed or if outliers are present, t procedures
are not robust.
b. For a sample size from 15 to 39: t procedures are robust
except in the presence of outliers or strong skewness.
c. For a sample size of 40 or above: t procedures are robust
even for clearly skewed distributions.
ii. Calculate the confidence interval.
1. Xbar ± t* x (sx / “square root n”)
2. Find t* from the appropriate row in Table C. You need to find the
row that is closes to the number of “degrees of freedom”, where
the degrees of freedom are n – 1. (n is the sample size)
d. Conclude
i. “We are C% confident that the mean … is between … and …” (Use the
range implied from the confidence interval calculated above.)