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Stats-Chapter 6 Notes
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I. Review Material (Day 1)
A) Activity: Where do I stand?
1.) Let’s create a dotplot showing the height distribution of all the students in this class.
2.) Calculate the 5-number summary, the mean and standard deviation:
Min=
Q1=
Mean=
Med=
Q3=
Max=
Standard Deviation =
3.) Count the number of students in the class that have heights less than or equal to your height.
4.) Now calculate the percent of students that have heights less than or equal to your height.
5.) Where does your height fall relative to the mean: above or below?
6.) How far above or below the mean is it?
7.) How many standard deviations above or below the mean is it?
This last number is the
that corresponds to your height.
B) Percentiles
Definition:
Example 1: Mrs. Munson is concerned about how her daughter’s height and weight compare with those of
other girls her age. She uses an online calculator to determine that her daughter is at the 87th
percentile for weight and the 67th percentile for height. Explain to Mrs. Munson what this means.
Example 2: Peter is a star runner on the track team, and Molly is one of the best sprinters on the swim
team. Both athletes qualify for the league championship meet based on their performance during the
regular season.
a) In the track championships, Peter records a time that would fall at the 80th percentile of all his race
times that season. But his performance places him at the 50th percentile in the league championship
meet. Explain how this is possible.
b) Molly swims a bit slowly for her in the league swim meet, recording a time that would fall at the 50 th
percentile of all her meet times that season. But her performance places Molly at the 80th percentile in
this event at the league meet. Explain how this could happen.
C) Standard Score or z-score:
From standard deviation, we can assign a piece of data a “Standard Score,” which allows us to compare
that piece of data to other pieces in the set. This is calculated with the following formula:
Because of the subtraction, it IS POSSIBLE to get a standard score that is negative, which means it is
to the left (or below) the mean. A standard score of 1 means you are 1 standard deviation above the
mean, 2 means 2 above, etc.
Example 1:
Calculate the standard scores for each x-value
1. x=2,
z=__________
if the mean of the data is 10 and
2. x=12,
z=__________
the standard deviation is 2.
3. x=9,
z=__________
4. x=15,
z=___________
Example 2: Sophia scored a 660 on the SAT Math test. Jim took the ACT Math test and he received a
26. SAT scores have a mean of 500 and a standard deviation of 100. ACT scores are normally
distributed with mean 18 and standard deviation 6. Assuming that both tests measure the same kind of
ability, who did better?
Example 3: Francine, who is 25 years old, has her bone density measured using DEXA (dual-energy X-ray
absorptiometry test). Her results indicate a bone density in the hip of 948 g/cm2, which converts to a
standardized score of z = -1.45, In the population of 25-year-old women like Francine, the mean bone
density in the hip is 956 g/cm2.
a) Francine has not taken a statistics class in a few years. Explain to her in simple language what the
standardized score tells her about her bone density.
b) Use the information provided to calculate the standard deviation of bone density in the reference
population.
c) Francine’s friend, Lisa, who is 35 years old, has her bone density measured using DEXA. Hers is
reported as 948 g/cm2, but her standardized score is z = 0.50. The mean bone density in the hip for the
reference population of 35-year-old women is 944 g/cm2. Whose bones are healthier- Francine’s or
Lisa’s? Justify your answer.
Table 3.1 shows the salaries for each member of the Phillies baseball team on the opening day of the
2008 season. Figure 3.2 gives a dotplot and summary statistics for the salary data. Use this
information to answer the questions below.
Brad Lidge played a crucial role as the Phillies’ “closer”; that is, he pitched the end of many games
throughout the season.
a) Find the percentile corresponding to Lidge’s salary. Explain what this value means.
b) Find the z-score corresponding to Lidge’s salary. Explain what this value means.
c) Did Ryan Madson have a high or low salary compared with the rest of the team? Justify your answer
using percentile and z-score.
II. Section 6.1 - The Normal Distribution (Day 2)
A) Density Curves
Definition: A density (also called probability) curve is a curve that


A density curve describes the overall pattern of a distribution. The area under the curve and above any
range of values is the proportion of all observations that fall in that range.
Example 1: The histogram represents the scores of 947 seventh-grade students in Gary, Indiana on the
Iowa Test of Basic Skills.
B) Normal Distributions
Just about anything you measure turns out to be normally distributed, or at least approximately so.
That is, usually most of the observations cluster around the mean, with progressively fewer observations
out towards the extremes
Although most variables are normally distributed, it is not the case that all variables are normally
distributed.
As examples, consider the following:
1) Values of a dice roll
2) Flipping a coin
3) Salary distributions
Make a sketch of each example below:
1)
2)
3)
The first two distributions are called
.
Normal curves are symmetric, single-peaked, and bell-shaped. All normal distributions have similar
shapes and are determined solely by their mean µ and standard deviation σ. The points at which the
curve changes concavity are located a distance σ on either side of µ. We will use the area under these
curves to represent a percentage of observations. (These areas correspond to integrals, for those of
you with some experience with calculus.)
The normal distribution has two parameters (two numerical descriptive measures)
1)
2)
If X is a quantity to be measured that has a normal distribution with mean (μ) and standard deviation (σ),
we designate this by writing
Properties of Normal Curves
1)
2)
3)
4)
5)
a. Sharply peaked and compact =
b. Low, wider peak =
6)
Example 1
Suppose X ~ N(5, 6).
This says that x is a normally distributed random variable with mean μ = 5 and standard deviation σ = 6.
a) Suppose x = 17. Then:
b) Now suppose x = 1
Try It
What is the z-score of x, when x = 1 and X~N(12,3)?
C) Applications of the Normal Curve
Three reasons we are interested in normal distributions:
1. Normal distributions are good descriptions for some distributions of real data. Examples include
test scores and characteristics of biological populations (such as height or weight).
2. Normal distributions are good approximations to the results of many kinds of chance outcomes.
An example is the proportion of heads in a repeatedly tossed coin experiment.
3. Many statistical inference procedures based on normal distributions work well for other roughly
symmetric distributions.
D) The Empirical Rule
Note. In the normal distribution with mean µ and standard deviation σ:



This is also called the “68-95-99.7 Rule.”
What percentiles do the quartiles
correspond to?
Q1
Q3
Example 1: The mean height of 15 to 18 year-old males from Chile from 2009 to 2010 was 170 cm
with a standard deviation of 6.28 cm. Male heights are known to follow a normal distribution.
Let X = the height of a 15 to 18-year-old male from Chile in 2009 to 2010. Then X~N(170, 6.28).
a. Suppose a 15 to 18-year-old male from Chile was 168 cm tall from 2009 to 2010.
The z-score when x = 168 cm is z =_______.
This z-score tells you that x =168 is________ standard deviations to the________(right or left)
of the mean _____ (What is the mean?).
b. Suppose that the height of a 15 to 18-year-old male from Chile from 2009 to 2010 has a zscore of z = 1.27. What is the male’s height?
The z-score (z=1.27) tells you that the male’s height is ________ standard deviations to the
__________ (right or left) of the mean.
Example 2: From 1984 to 1985, the mean height of 15 to 18-year-old males from Chile was 172.36 cm,
and the standard deviation was 6.34 cm.
Let Y= the height of 15 to 18-year-old males in 1984 to 1985. Then Y~N(172.36, 6.34).
a. About 68% of the y values lie between what two values?
These values are________________.
The z-scores are ________________, respectively.
b. About 95% of the y values lie between what two values?
These values are________________.
The z-scores are ________________ respectively.
c. About 99.7% of the y values lie between what two values?
These values are ________________.
The z scores are ________________, respectively.
Example 3: The scores on a college entrance exam have an approximate normal distribution with mean,
µ = 52 points and a standard deviation, σ= 11 points.
a. About 68% of the y values lie between what two values?
These values are ________________.
The z-scores are ________________, respectively.
b. About 95% of the y values lie between what two values?
These values are ________________.
The z-scores are ________________, respectively.
c. About 99.7% of the y values lie between what two values?
These values are ________________.
The z-scores are ________________, respectively.
Example 4: Given the following stemplot that gives IQ test scores for 74 seventh grade students…
a) Between what values do the IQ scores of 95% of all seventh-graders lie? Explain how you found your
answer.
b) What percent of IQ scores for seventh-graders are above 100? Explain.
c) What percent of all students have IQ scores of 144 or higher? None of the 74 students in our
sample had scores this high. Does this surprise you? Why or why not?
Day 4: Complete 6.1 Practice Worksheet
III. Section 6.2: Using the Standard Normal Distribution and z-scores (Day 4)
A) Using the Calculator
All of the problems we just calculated can also be done on the calculator.
Follow the following steps to find:
1) the percent of observations in a specific range2nd
VARS [DISTR} 2: :normalcdf
Enter the following info into normalcdf (lower bound, upper bound, µ., σ)
2) score needed to achieve specific percentile2nd
VARS [DISTR} 3 : invnorm
Enter the following info into the calculator: invnorm (percentile (as a decimal), µ., σ)
Practice: Using the SAT data of µ = 500 and σ = 100, use the calculator to find each of the following.
Write down the numbers you are entering into the calculator:
a) % scoring lower than 720
b) % scoring lower than 400
c) % scoring higher than 580
d) score needed to be in the top 5%
e) score needed to be in the top 25%
f) score needed to be in the top 37%
g) % of scores between 500 and 800
h) % of scores between 550 and 650
i) % of scores between 500 and 600
j) % of scores between 300 and 700
B.) Normal Distribution Practice
Show all work including calculator entries
1.) On the driving range, Tiger Woods practices his swing with a particular club by hitting many, many
balls. When Tiger hits the ball, the distance the ball travels follows a normal distribution with mean 304
yards and standard deviation 8 yards. What percent of Tiger’s drives travel at least 290 yards? Be
sure to sketch the curve first.
2.) What percent of Tiger’s drives travel between 305 and 325 yards?
3.) What distance would a ball have to travel to be at the 80th percentile of Tiger’s drive lengths?
4.) Scores on the Wechsler Adult Intelligence Scale for 20 to 34-year-olds are approximately normally
distributed with mean 110 and standard deviation 25. How high must a person score to be in the top
25% of all scores?
5.) The average performance of females on the SAT, especially the Math section, is lower than that of
males. The reasons for this gender gap are controversial. In 2007, female scores on the SAT Math test
followed a normal distribution with mean 500 and standard deviation 111. Male scores had a mean of 533
and a standard deviation of 118. What percent of females scored higher than the male mean?
6.) Find the 85th percentile of the SAT Math score distribution for males.
7.) To what percentile in the female score distribution does your answer to the previous question
correspond?
8.) Bags of potatoes in a shipment averaged 10 pounds with a standard deviation of 0.5 pounds. A
histogram of these weights followed a normal curve quite closely.
a) What percent of the bags weighed less than 10.25 pounds?
b) What percent weighed between 9.5 and 10.25 pounds?