1. Chapter 2: Modeling Distributions of Data
Section 2.2
Normal Distributions
The Practice of Statistics, 4th edition - For AP*
STARNES, YATES, MOORE

2. Chapter 2 Modeling Distributions of Data
2.1 Describing Location in a Distribution
2.2 Normal Distributions

3. Section 2.2 Normal Distributions
Learning Objectives
After this section, you should be able to…
DESCRIBE and APPLY the Rule
DESCRIBE the standard Normal Distribution
PERFORM Normal distribution calculations
ASSESS Normality

4. In 1995 the Educational Testing Service (ETS) adjusted the scores of SAT tests. Before ETS re-centered the SAT verbal test, the mean of all test scores was 450.
1. How would adding 50 points to each score affect the mean?
2. The standard deviation was 100 points. What would the standard deviation be after adding 50 points?
WARM UP

5. In 1995 the Educational Testing Service (ETS) adjusted the scores of SAT tests. Before ETS re-centered the SAT verbal test, the mean of all test scores was 450.
1. How would adding 50 points to each score affect the mean? The new mean would be 500.
2. The standard deviation was 100 points. What would the standard deviation be after adding 50 points? The standard deviation would remain unchanged. It would still be 100.
WARM UP

6. WARM UP
Use the cumulative relative frequency graph to answer the following questions.
What value is the
20th percentile?
2. Find the median.
3. Find the IQR.

7. WARM UP
2000 freshmen at State University took a biology test. The scores were distributed normally with a mean of 70 and a standard deviation of 5.
1. What percentage of scores are between scores 65 and 75?
2. What percentage of scores are between scores 60 and 85?
3. Approximately how many biology students scored between 55 and 60?

8. WARM UP
2000 freshmen at State University took a biology test. The scores were distributed normally with a mean of 70 and a standard deviation of 5.
1. What percentage of scores are between scores 65 and 75? 68%
2. What percentage of scores are between scores 60 and 75? %
3. Approximately how many students scored between 60 and 65? 270

9. Describing Location in a Distribution
Density Curves
In Chapter 1, we developed a kit of graphical and numerical tools for describing distributions. Now, we’ll add one more step to the strategy.
So far our strategy for exploring data is :
1. Graph the data to get an idea of the overall pattern
2. Calculate an appropriate numerical summary to describe the center and spread of the distribution.
Sometimes the overall pattern of a large number of observations is so regular, that we can describe it by a smooth curve, called a density curve.
Describing Location in a Distribution

10. Describing Location in a Distribution
Density Curve
Definition:
A density curve is a curve that
is always on or above the horizontal axis, and
has area exactly 1 underneath it.
A density curve describes the overall pattern of a distribution. The area under the curve and above any interval of values on the horizontal axis is the proportion of all observations that fall in that interval.
Describing Location in a Distribution
The overall pattern of this histogram of the scores of all 947 seventh-grade students in Gary, Indiana, on the vocabulary part of the Iowa Test of Basic Skills (ITBS) can be described by a smooth curve drawn through the tops of the bars.

11. Here’s the idea behind density curves:

12. Normal Distributions Normal Distributions
One particularly important class of density curves are the Normal curves, which describe Normal distributions.
All Normal curves are symmetric, single-peaked, and bell- shaped
A Specific Normal curve is described by giving its mean µ and standard deviation σ.
Normal Distributions
Two Normal curves, showing the mean µ and standard deviation σ.

13. Normal Distributions Normal Distributions Definition:
A Normal distribution is described by a Normal density curve. Any particular Normal distribution is based on two numbers: its mean µ and standard deviation σ.
The mean of a Normal distribution is the center of the symmetric Normal curve.
The standard deviation is the distance from the center to the change-of-curvature points on either side.
We abbreviate the Normal distribution with mean µ and standard deviation σ as N(µ,σ).
Normal distributions are good descriptions for some distributions of real data.
Normal distributions are good approximations of the results of many kinds of chance outcomes.

14. Normal Distributions The 68-95-99.7 Rule
Although there are many Normal curves, they all have properties in common.
Normal Distributions
Definition: The Rule (“The Empirical Rule”)
In the Normal distribution with mean µ and standard deviation σ:
Approximately 68% of the observations fall within σ of µ.
Approximately 95% of the observations fall within 2σ of µ.
Approximately 99.7% of the observations fall within 3σ of µ.

15. The distribution of Iowa Test of Basic Skills (ITBS) vocabulary scores for 7th grade students in Gary, Indiana, is close to Normal. Suppose the distribution is N(6.84, 1.55).
Sketch the Normal density curve for this distribution.
What percent of ITBS vocabulary scores are less than 3.74?
What percent of the scores are between 5.29 and 9.94?
Example, p. 113
Normal Distributions

16. Suppose it takes you 20 minutes, on average, to drive to school, with a standard deviation of 2 minutes. Suppose a Normal model is appropriate for the distribution of driving times.
1. What percentage of times will you arrive at school in less than 22 minutes?
2. What percentage of times will it take you more than 24 minutes?
3. In the 2006 Winter Olympics, Jean-Baptiste Grange of France skied the slalom in seconds – 1 standard deviation faster than the mean. If the slalom times follow a Normal distribution, about how many of the 35 skiers would you expect skied the slalom faster than Jean- Baptiste?
WARM UP

17. The heights of men are Normally distributed with a mean of 69 inches and a standard deviation of 2.5 inches.
1. Draw and label a Normal curve with this mean and standard deviation.
2. What percent of men are taller than 74 inches?
3. Between what heights do the middle 95% of men fall?
4. What percent of men are between 64 and inches tall?
5. A height of 71.5 inches corresponds to what percentile?
WARM UP

18. The heights of men are Normally distributed with a mean of 69 inches and a standard deviation of 2.5 inches.
1. Draw and label a Normal curve with this mean and standard deviation.
2. What percent of men are taller than 74 inches? 2.5%
3. Between what heights do the middle 95% of men fall?
Between 64 and 74 inches
4. What percent of men are between 64 and 66.5 inches tall? 13.5%
5. A height of 71.5 inches corresponds to what percentile? 84th percentile
WARM UP – Answers

19. The Standard Normal Distribution
All Normal distributions are the same if we measure in units of size σ from the mean µ as center.
Normal Distributions
Definition:
The standard Normal distribution is the Normal distribution with mean 0 and standard deviation 1.
If a variable x has any Normal distribution N(µ,σ) with mean µ and standard deviation σ, then the standardized variable
has the standard Normal distribution, N(0,1).

20. The Standard Normal Table
Normal Distributions
Because all Normal distributions are the same when we standardize, we can find areas under any Normal curve from a single table.
Definition: The Standard Normal Table
Table A is a table of areas under the standard Normal curve. The table entry for each value z is the area under the curve to the left of z.
Suppose we want to find the proportion of observations from the standard Normal distribution that are less than 0.81.
We can use Table A:
P(z < 0.81) =
.7910 Z
.00
.01
.02
0.7
.7580
.7611
.7642
0.8
.7881
.7910
.7939
0.9
.8159
.8186
.8212

21. Find the proportion of observations from the standard normal distribution that are…
1. less than 1.39
2. greater than -2.15
(Notice now you are finding the area to the RIGHT of -2.15)
Standard Normal Table

22. Normal Distributions Finding Areas Under the Standard Normal Curve
Example, p. 117
Normal Distributions
Find the proportion of observations from the standard Normal distribution that are between and 0.81.
– =

23. Standard Normal Table Try this one on your own!
Find the proportion of observations from the standard normal distribution that are…
Between and 1.81
Standard Normal Table

24. Standard Normal Distribution
Use Table A in the back of your book to find the value of z from the standard Normal distribution that satisfies each of the following conditions. Sketch a standard Normal curve and shade the area representing the region.
e) The 20th percentile
f) 45% of all observations are greater than z
Standard Normal Distribution

25. Standard Normal Distribution
Example: Percentiles and z-scores
A distribution of test scores is approximately Normal and Joe scores in the 85th percentile. How many standard deviations above the mean did he score?
In a Normal distribution, Q1 is how many standard deviations below the mean?
Standard Normal Distribution

26. Normal Distribution Calculations
Normal Distributions
How to Solve Problems Involving Normal Distributions
State: Express the problem in terms of the observed variable x.
Plan: Draw a picture of the distribution and shade the area of interest under the curve.
Do: Perform calculations.
Standardize x to restate the problem in terms of a standard Normal variable z.
Use Table A and the fact that the total area under the curve is 1 to find the required area under the standard Normal curve.
Conclude: Write your conclusion in the context of the problem.

27. Normal Distribution Calculations
Normal Distributions
In the 2008 Wimbledon tennis tournament, Rafael Nadal averaged 115 miles per hour on his first serves. Assume the distribution of his first speeds follows the distribution N(115, 6). About what proportion of his first serves would you expect to exceed 120 mph?
State: Let x = the speed of Nadal’s first serve. We want to find the proportion of first serves with x > 120
Plan: Sketch the normal distribution and shade the area of interest
Do: Standardize 120 mph (turn it into a z-score)
Look up the z-score in the table
Conclude: We expect about 20% of Nadal’s first serves will travel more than 120 mph.

28. Normal Distribution Calculations
Normal Distributions
In the 2008 Wimbledon tennis tournament, Rafael Nadal averaged 115 miles per hour on his first serves. Assume the distribution of his first speeds follows the distribution N(115, 6).
What percent Rafael Nadal’s first serves are between 100 and 110 mph?
State:
Plan:
Do:
Conclude:

29. Normal Distribution Calculations
Normal Distributions
Example: According to the CDC, the height of 3 year old females are approximately normally distributed with a mean of 94.5 cm and a standard deviation of 4 cm.
(a) What percent of 3 year old females are taller than 100 cm?
State:
Plan:
Do:
Conclude:

30. Normal Distribution Calculations
Normal Distributions
Example: According to the CDC, the height of 3 year old females are approximately normally distributed with a mean of 94.5 cm and a standard deviation of 4 cm.
(b) What percent of 3 year old females are between 90 and 95 cm?
State:
Plan:
Do:
Conclude:

31. WARM UP
Find the proportion of values in the standard Normal distribution for each of the following.
To the left of z = 1.23
To the right of z = -0.85
Between z = and z = 0.75
Find the z-value that corresponds to the following:
4. The 28th percentile
5. The 70th percentile
6. The median of a distribution

32. WARM UP – Answers
Find the proportion of values in the standard Normal distribution for each of the following.
To the left of z =
To the right of z =
Between z = and z =
Find the z-value that corresponds to the following:
4. The 28th percentile
5. The 70th percentile
6. The median of a distribution 0

33. Normal Distribution Calculations
Normal Distributions
How do you do Normal calculations on the calculator? What do you need to show on the AP exam?
Suppose that Zach Greinke of the Kansas City Royals throws his fastball with a mean velocity of 94 miles per hour (mph) and a standard deviation of 2 mph and that the distribution of his fastball speeds is can be modeled by a Normal distribution.
(a) About what proportion of his fastballs will travel over 100 mph?
   
(b) About what proportion of his fastballs will travel less than 90 mph?   
(c) About what proportion of his fastballs will travel between 93 and 95 mph?
(d) What is the 30th percentile of Greinke’s distribution of fastball velocities?

34. Assessing Normality Normal Distributions
The Normal distributions provide good models for some distributions of real data. Many statistical inference procedures are based on the assumption that the population is approximately Normally distributed. Consequently, we need a strategy for assessing Normality.
Normal Distributions
1. Plot the data.
Make a dotplot, stemplot, or histogram and see if the graph is approximately symmetric and bell-shaped.
2. Check whether the data follow the rule.
Count how many observations fall within one, two, and three standard deviations of the mean and check to see if these percents are close to the 68%, 95%, and 99.7% targets for a Normal distribution.

35. Normal Probability Plots
Most software packages can construct Normal probability plots. These plots are constructed by plotting each observation in a data set against its corresponding percentile’s z-score.
Normal Distributions
Interpreting Normal Probability Plots
If the points on a Normal probability plot lie close to a straight line, the plot indicates that the data are Normal. Systematic deviations from a straight line indicate a non-Normal distribution. Outliers appear as points that are far away from the overall pattern of the plot.

36. A cereal manufacturer has a machine that fills the boxes
A cereal manufacturer has a machine that fills the boxes. Based on their experience with the machine, the company believes that the amount of cereal in the boxes follows N(16.3, 0.2).
1. If the company claims that a box of cereal weighs16 ounces, what percent of the boxes are underweight?
SAT Verbal scores follow a Normal distribution with a mean of 500 and a standard deviation of 100.
2. What proportion of scores fall between 450 and 600?
3. Suppose a college says it admits only people with SAT Verbal scores among the top 10%. How high of a score does it take to be eligible?
WARM UP

37. A cereal manufacturer has a machine that fills the boxes
A cereal manufacturer has a machine that fills the boxes. Based on their experience with the machine, the company believes that the amount of cereal in the boxes follows N(16.3, 0.2).
1. If the company claims that a box of cereal weighs16 ounces, what percent of the boxes are underweight?
SAT Verbal scores follow a Normal distribution with a mean of 500 and a standard deviation of 100.
2. What proportion of scores fall between 450 and 600?
3. Suppose a college says it admits only people with SAT Verbal scores among the top 10%. How high of a score does it take to be eligible? 630
WARM UP – Answers

38. Assume the cholesterol levels of adult American women can me described by a Normal model with a mean of 188 mg/dL and a standard deviation of 24.
1. What percent of adult women do you expect to have cholesterol levels over 200?
2. What percent of adult women do you expect to have levels between 150 and 170?
3. Find the 35% percentile of the distribution.
4. Estimate the IQR of the distribution.
WARM UP

39. Assume the cholesterol levels of adult American women can me described by a Normal model with a mean of 188 mg/dL and a standard deviation of 24.
1. What percent of adult women do you expect to have cholesterol levels over 200?
2. What percent of adult women do you expect to have levels between 150 and 170?
3. Find the 35% percentile of the distribution
4. Estimate the IQR of the distribution
WARM UP – Answers

40. Section 2.2 Normal Distributions
Summary
In this section, we learned that…
The Normal Distributions are described by a special family of bell- shaped, symmetric density curves called Normal curves. The mean µ and standard deviation σ completely specify a Normal distribution N(µ,σ). The mean is the center of the curve, and σ is the distance from µ to the change-of-curvature points on either side.
All Normal distributions obey the Rule, which describes what percent of observations lie within one, two, and three standard deviations of the mean.

41. Section 2.2 Normal Distributions
Summary
In this section, we learned that…
All Normal distributions are the same when measurements are standardized. The standard Normal distribution has mean µ=0 and standard deviation σ=1.
Table A gives percentiles for the standard Normal curve. By standardizing, we can use Table A to determine the percentile for a given z-score or the z-score corresponding to a given percentile in any Normal distribution.
To assess Normality for a given set of data, we first observe its shape. We then check how well the data fits the rule. We can also construct and interpret a Normal probability plot.

42. Looking Ahead… In the next Chapter…
We’ll learn how to describe relationships between two quantitative variables
We’ll study
Scatterplots and correlation
Least-squares regression
In the next Chapter…