Chapter 2 Review
Group Members:
Aditya “dits” Shah
Alex “wu-tang” Wu
Akash “try-hard” Doshi
The Big Idea
• A variable’s location in a distribution can be
described by its standardized form or
percentile. Many real life situations, such as
grades and population growth, follow a type
of distribution called Normal distribution.
Finding a variable’s location in this distribution
is the most important part of this chapter.
Vocabulary You Need to Know
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Density curve
Normal distribution
Normal curve
Z-score
Standardization
Percentile
Normal probability plot
Density Curves
• These curves give us a mathematical model for distributions
• They are either on or above the horizontal axis and have an area of
1 underneath
• For density curves our median is the equal areas point and the
mean is the “balance point”
• We can tell a curve is skewed if the mean is pulled towards the tail
Normal Distribution
• Normal distribution is classified as a continuous, symmetrical,
bell-shaped distribution of a variable
• Other characteristics are that the mean, median, and mode
are all equal and located at the middle of the curve.
• The curve is unimodal (one mode) and it never touches the xaxis (the axis acts like an asymptote)
• AREA UNDER THE CURVE IS EQUAL TO 1!!
Empirical Rule
• The 68-95-99.7 rule
• 68% of all data lies within 1 standard dev. from the
mean. 95% of all data lies within 2 standard dev.
from the mean. 99.7% of all data lies within 3
standard dev. from the mean.
• We use empirical rule to predict outcomes, estimate
impending data, etc.
Z-Score
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Z=
We use z scores to standardize x values
If z is negative, use the table on the left-hand page
If z is positive, use the table on the right-hand page
When working backwards, find the “p” value you are
given on the table in your book/given. If we say our
p=.95 we look for where a z-score equals this. In this
case it is halfway between .9495 and .9505, so we
average the z-scores 1.64 and 1.65 to get a final z-score
of 1.645.
Normal Probability Plot
• This is a graphical technique for normality testing-whether or not a
data set is normally distributed
• The data are plotted against a theoretical normal distribution in
such a way that the points should form an approximate straight
line. Deviations from this straight line indicate departures from
normality.
Formulas You Should Know
• z=
• Empirical rule:
- Approximately 68% of observations fall
within 1σ from the mean
- Approximately 95% of observations fall
within 2σ from the mean
- Approximately 99.7% of observations fall
within 3σ from the mean
• z-score conversion to proportion/percentile
Calculator Key Strokes
• 1st keystroke:
- The normal cumulative distribution function
- This finds the area under a normal curve up to some
standardized point
- Press 2nd -> Vars -> normalcdf(
- Enter the lower bound, upper bound, mean, and standard
deviation in that order (math print will facilitate this lawlz)
- Hit enter and see your area
• 2nd keystroke:
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The inverse normal distribution function
This finds the z-score of some upper bound for a given area
Press 2nd -> Vars -> invnorm
Enter the area under the curve, mean, and standard deviation
Hit enter and see your area
Example Problem(s)
• The weights of packages of ground beef are normally distributed d s
bu ed with mean 1 pound and standard deviation .10. What is the
probability that a randomly selected package weighs between 0.80
and 0.85 pounds?
• A Company produces “20 ounce” bottles of water. The true
amounts of water in the bottles of this brand follow a normal
distribution. Suppose the companies “20 ounce” bottles follow a
normally distribution with a mean μ=20.2 ounces with a standard
deviation σ=0.125 ounces. What proportion of the jars are
under‐filled (i.e.,have less than 20 ounces of water)?
• What proportion of the water bottles contain between 20 and 20.3
ounces of water.
• 99% of the bottles of water will contain more than what amount of
water?
Question 1
Question 2
Question 3
Question 4
Helpful Hints
• For empirical rule: before applying the rule it’s a
good idea to identify the data being described
and the mean and standard dev. ALWAYS SKETCH
A GRAPH with all the information to help visualize
the scenario. CHECK TO MAKE SURE YOU CAN
USE THE EMPIRACAL RULE (remember the
characteristics!!)
• Steps: State the problem, use the table, draw and
shade a picture, and unstandardize to obtain your
x