Statistics 111 - Lecture 18
Final review
June 2, 2008
Stat 111 - Lecture 18 - Review
1
Administrative Notes
• Pass in homework
• Final exam on Thursday
• Office hours today from 12:30-2:00pm
June 2, 2008
Stat 111 - Lecture 18 - Review
2
Final Exam
• Tomorrow from 10:40-12:10
• Show up a bit early, if you arrive after
10:40 you will be wasting time for your
exam
• Calculators are definitely needed!
• Single 8.5 x 11 cheat sheet (two-sided)
allowed
• Sample midterm is posted on the website
• Solutions for the midterm are posted on
the website
June 2, 2008
Stat 111 - Lecture 18 - Review
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Final Exam
• The test is going to be long – 6 or 7 questions
• Pace yourself
• Do NOT leave anything blank!
• I want to give you credit for what you know
give me an excuse to give you points!
• The questions are NOT in order of difficulty
• If you get stuck on a question put it on hold
and try a different question
June 2, 2008
Stat 111 - Lecture 18 - Review
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Outline
•
•
•
•
•
•
•
•
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Collecting Data (Chapter 3)
Exploring Data - One variable (Chapter 1)
Exploring Data - Two variables (Chapter 2)
Probability (Chapter 4)
Sampling Distributions (Chapter 5)
Introduction to Inference (Chapter 6)
Inference for Population Means (Chapter 7)
Inference for Population Proportions (Chapter 8)
Inference for Regression (Chapter 10)
June 2, 2008
Stat 111 - Lecture 18 - Review
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Experiments
Treatment Group
Population
•
•
•
•
•
Experimental
Units
Control Group
Treatment
No Treatment
Try to establish the causal effect of a treatment
Key is reducing presence of confounding variables
Matching: ensure treatment/control groups are very
similar on observed variables eg. race, gender, age
Randomization: randomly dividing into treatment or
control leads to groups that are similar on observed
and unobserved confounding variables
Double-Blinding: both subjects and evaluators don’t
know who is in treatment group vs. control group
June 2, 2008
Stat 111 - Lecture 18 - Review
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Sampling and Surveys
Population
?
Parameter
Sampling
Sample
Inference
Estimation
Statistic
• Just like in experiments, we must be cautious of
potential sources of bias in our sampling results
• Voluntary response samples, undercoverage, nonresponse, untrue-response, wording of questions
• Simple Random Sampling: less biased since each
individual in the population has an equal chance of
being included in the sample
June 2, 2008
Stat 111 - Lecture 18 - Review
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Different Types of Graphs
• A distribution describes what values a variable
takes and how frequently these values occur
• Boxplots are good for center,spread, and outliers
but don’t indicate shape of a distribution
• Histograms much more effective at displaying the
shape of a distribution
June 2, 2008
Stat 111 - Lecture 18 - Review
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Measures of Center and Spread
• Center: Mean
𝑥𝑖 𝑥1 + 𝑥2 + ⋯ + 𝑥𝑛
𝑋=
=
𝑛
𝑛
• Spread: Standard Deviation
𝑠=
(𝑥𝑖 − 𝑥)2
𝑛−1
• For outliers or asymmetry, median/IQR are better
• Center: Median - “middle number in distribution”
• Spread: Inter-Quartile Range IQR = Q3 - Q1
• We use mean and SD more since most distributions
we use are symmetric with no outliers(eg. Normal)
June 2, 2008
Stat 111 - Lecture 18 - Review
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Relationships between continuous var.
• Scatterplot examines relationship between response
variable (Y) and a explanatory variable (X):
Education and Mortality:
r = -0.51
• Positive vs. negativeassociations
• Correlation is a measure of the strength of linear
relationship between variables X and Y
• r near 1 or -1 means strong linear relationship
• r near 0 means weak linear relationship
• Linear Regression: come back to later…
June 2, 2008
Stat 111 - Lecture 18 - Review
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Probability
• Random process: outcome not known exactly, but
have probability distribution of possible outcomes
• Event: outcome of random process with prob. P(A)
• Probability calculations: combinations of rules
•
•
•
•
Equally likely outcomes rule
Complement rule
Additive rule for disjoint events
Multiplication rule for independent events
• Random variable: a numerical outcome or summary
of a random process
• Discrete r.v. has a finite number of distinct values
• Continuous r.v. has a non-countable number of values
• Linear transformations of variables
June 2, 2008
Stat 111 - Lecture 18 - Review
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The Normal Distribution
• The Normal distribution has center  and spread 2
N(0,1)
N(-1,2)
N(0,2)
N(2,1)
• Have tables for any probability from the standard
normal distribution ( = 0 and 2 = 1)
• Standardization: converting X which has a N(,2)
distribution to Z which has a N(0,1) distribution:
𝑋−𝜇
𝑍=
𝜎
• Reverse standardization: converting a standard
normal Z into a non-standard normal X
𝜎𝑍 + 𝜇 = 𝑋
June 2, 2008
Stat 111 - Lecture 18 - Review
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Inference using Samples
Population
?
Parameters:  or p
Sampling
Sample
Inference
Estimation
Statistics: 𝑋 or 𝑝
• Continuous: pop. mean estimated by sample mean
• Discrete: pop. proportion estimated by sample proportion
• Key for inference: Sampling Distributions
• Distribution of values taken by statistic in all possible samples
from the same population
June 2, 2008
Stat 111 - Lecture 18 - Review
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Sampling Distribution of Sample Mean
• The center of the sampling distribution of the sample
mean is the population mean: mean(𝑋) = 𝜇
• Over all samples, the sample mean will, on average, be
equal to the population mean (no guarantees for 1 sample!)
• The standard deviation of the sampling distribution
𝜎
of the sample mean is
𝑆𝐷 𝑋 =
𝑛
• As sample size increases, standard deviation of the sample
mean decreases!
• Central Limit Theorem: if the sample size is large
enough, then the sample mean 𝑋 has an
approximately Normal distribution
June 2, 2008
Stat 111 - Lecture 18 - Review
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Binomial/Normal Dist. For Proportions
• Sample count Y follows Binomial distribution which
we can calculate from Binomial tables in small samples
• If the sample size is large (n·p and n(1-p)≥ 10), sample
count Y follows a Normal distribution:
mean(𝑌) = 𝑛𝑝
𝑆𝐷 𝑌 =
𝑛𝑝(1 − 𝑝)
• If the sample size is large, the sample proportion also
approximately follows a Normal distribution:
mean(𝑝) = 𝑝
𝑆𝐷 𝑝 =
June 2, 2008
𝑝(1 − 𝑝)
𝑛
Stat 111 - Lecture 18 - Review
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Summary of Sampling Distribution
Type of
Data
Continuous
Unknown
Parameter

Statistic
𝑋
Variability of
Statistic
𝑆𝐷 𝑋 =
𝜎
𝑛
Distribution
of Statistic
Normal
(if n large)
Binomal
(if n small)
Count
Xi = 0 or 1
June 2, 2008
p
𝑝
𝑆𝐷 𝑝 =
𝑝(1 − 𝑝)
𝑛
Normal
(if n large)
Stat 111 - Lecture 18 - Review
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Introduction to Inference
• Use sample estimate as center of a confidenceinterval
of likely values for population parameter
• All confidence intervals have the same form:
Estimate ± Margin of Error
• The margin of error is always some multiple of the
standard deviation (or standard error) of statistic
• Hypothesis test: data supports specific hypothesis?
1. Formulate your Null and Alternative Hypotheses
2. Calculate the test statistic: difference between data
and your null hypothesis
3. Find the p-value for the test statistic: how probable is
your data if the null hypothesis is true?
June 2, 2008
Stat 111 - Lecture 18 - Review
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Inference: Single Population Mean 
• Known SD : confidence intervals and test statistics
involve standard deviation and normal critical values
𝑋−𝑍
∗
𝜎
𝑛
,𝑋 + 𝑍
∗
𝜎
𝑇=
𝑛
𝑋 − 𝜇0
𝜎
𝑛
• Unknown SD : confidence intervals and test
statistics involve standard error and critical values
from a t distribution with n-1 degrees of freedom
𝑋−
∗
𝑡𝑛−1
𝑠
𝑛
,𝑋 +
∗
𝑡𝑛−1
𝑠
𝑛
𝑇=
𝑋 − 𝜇0
𝑠
𝑛
• t distribution has wider tails (more conservative)
June 2, 2008
Stat 111 - Lecture 18 - Review
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Inference: Comparing Means 1and 2
• Known 1 and2: two-sample Z statistic uses normal
distribution
𝑋1 − 𝑋2
𝑍=
𝜎12 𝜎22
+
𝑛1 𝑛2
• Unknown 1 and2: two-sample T statistic uses t
distribution with degrees of freedom = min(n1-1,n2-1)
𝑋1 − 𝑋2
𝑇=
𝑠12 𝑠22
𝑠12 𝑠22
∗
∗
𝑋1 − 𝑋2 − 𝑡𝑘
+ , 𝑋 − 𝑋2 + 𝑡𝑘
+
𝑠12 𝑠22
𝑛1 𝑛2 1
𝑛1 𝑛2
+
𝑛1 𝑛2
• Matched pairs: instead of difference of two samples
X1 and X2, do a one-sample test on the difference d
𝑇=
June 2, 2008
𝑋𝑑 − 0
𝑠𝑑
𝑛
𝑋𝑑 −
∗
𝑡𝑛−1
Stat 111 - Lecture 18 - Review
𝑠𝑑
𝑛
, 𝑋𝑑 +
∗
𝑡𝑛−1
𝑠𝑑
𝑛
19
Inference: Population Proportion p
• Confidence interval for p uses the Normal
distribution and the sample proportion:
𝑝−𝑍
∗
𝑝(1 − 𝑝)
𝑝(1 − 𝑝)
∗
,𝑝 + 𝑍
𝑛
𝑛
𝑌
where 𝑝 =
𝑛
• Hypothesis test for p = p0 also uses the Normal
distribution and the sample proportion:
𝑝 − 𝑝0
𝑍=
𝑝0 (1 − 𝑝0 )
𝑛
June 2, 2008
Stat 111 - Lecture 18 - Review
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Inference: Comparing Proportions p1and p2
• Hypothesis test for p1 - p2 = 0 uses Normal
distribution and complicated test statistic
𝑌1
𝑌2
𝑝1 − 𝑝2
where 𝑝1 =
and 𝑝2 =
𝑍=
𝑛1
𝑛2
𝑆𝐸(𝑝1 − 𝑝2 )
with pooled standard error:
𝑌1 + 𝑌2
𝑆𝐸 𝑝1 − 𝑝2 = 𝑝𝑝 (1 − 𝑝𝑝 )
+
where 𝑝𝑝 =
𝑛1 𝑛2
𝑛1 + 𝑛2
• Confidence interval for p1 = p2 also uses Normal
distribution and sample proportions
1
𝑝1 − 𝑝2 ∓ 𝑍
June 2, 2008
∗
1
𝑝1 (1 − 𝑝1 ) 𝑝2 (1 − 𝑝2 )
+
𝑛1
𝑛2
Stat 111 - Lecture 18 - Review
21
Linear Regression
• Use best fit line to summarize linear relationship
between two continuous variables X and Y:
𝑌𝑖 = 𝛼 + 𝛽𝑋𝑖
• The slope (𝑏 = 𝑟 ∙ 𝑠𝑦 𝑠𝑥): average change you get in
the Y variable if you increased the X variable by one
• The intercept( 𝑎 = 𝑌 − 𝑏𝑋 ):average value of the Y
variable when the X variable is equal to zero
• Linear equation can be used to predict response
variable Y for a value of our explanatory variable X
June 2, 2008
Stat 111 - Lecture 18 - Review
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Significance in Linear Regression
• Does the regression line show a significant linear
relationship between the two variables?
H0 : = 0
versus Ha :  0
• Uses the t distribution with n-2 degrees of freedom
and a test statistic calculated from JMP output
𝑏
𝑇=
𝑆𝐸(𝑏)
• Can also calculate confidence intervals using JMP
output and t distribution with n-2 degrees of freedom
∗
𝑏 ∓ 𝑡𝑛−2
𝑆𝐸(𝑏)
June 2, 2008
∗
𝑎 ∓ 𝑡𝑛−2
𝑆𝐸(𝑎)
Stat 111 - Lecture 18 - Review
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Good Luck on Final!
• I have office hours until 2pm today
• Might be able to meet at other times
June 2, 2008
Stat 111 - Lecture 18 - Review
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