Download Chapters 1-3 Review KEY

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Chapters 1-3 Test Review Guide
KEY
The test will consist of both multiple choice and open-response questions, and the questions
will be AP style questions. The following is a list of concepts you need to know, along with
sample problems for select concepts.
a) Understand which graphs are appropriate for which variables (for example, a pie chart or bar
graph would be appropriate for categorical variables but not quantitative, just like a histogram
would be appropriate for quantitative variables but not categorical).
b) Read and interpret a stemplot, including determine the shape of a distribution from a
stemplot.
c) Understand the impact on mean and standard deviation of adding a particular value to a
distribution.
d) Know the difference between 𝜎 and 𝑠𝑥 and between 𝜇 and 𝑥̅ .
e) Test for outliers, create boxplots, and compare distributions from the boxplots.
f) Use the 68-95-99.7 rule to find boundaries.
g) Calculate Normal probabilities (above a value, below a value, between two values).
h) Work backwards in the Normal distribution to find values if given an area or percentile.
i) Work backwards in the Normal distribution to calculate standard deviation.
j) Make predictions given a least-squares regression line.
k) Calculate residuals.
l) Utilize regression computer output to calculate the equation of the least-squares regression
line and to calculate correlation.
m) Understand what information the residual plot provides.
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Sample Problems
1a) A distribution has a mean of 10 and a standard deviation of 4. If a score of 12 was added to
the distribution, what would happen to the mean and standard deviation?
Mean will increase (12 is larger than the mean of 10)
Standard deviation will decrease (12 is within one standard deviation of the mean)
b) Instead of a score of 12, a score of 3 was added to the distribution. What would happen to
the mean and standard deviation?
Mean will decrease (3 is smaller than the mean of 10)
Standard deviation will increase (3 is outside of one standard deviation of the mean)
2) Suppose that the distance a golfer can hit the ball has an approximately Normal distribution
with a mean of 150 yards and a standard deviation of 10 yards.
a) The middle 95% of his hits will travel between what two distances?
Use the 68-95-99.7 rule.
𝟏𝟓𝟎 − 𝟏𝟎 − 𝟏𝟎 = 𝟏𝟑𝟎
𝟏𝟓𝟎 + 𝟏𝟎 + 𝟏𝟎 = 𝟏𝟕𝟎
The middle 95% of his hits will travel between 130 and 170 yards.
b) The highest 15% of his hits will travel at least what distance?
𝑵(𝟏𝟓𝟎, 𝟏𝟎)
𝐢𝐧𝐯𝐍𝐨𝐫𝐦(𝐚𝐫𝐞𝐚: 𝟎. 𝟖𝟓, 𝝁: 𝟏𝟓𝟎, 𝝈: 𝟏𝟎) = 𝟏𝟔𝟎. 𝟑𝟔𝟒𝟑
The highest 15% of his hits will travel at least 160.36 yards.
3) For a group of college women, the association between height and weight is roughly linear.
The equation of the least-squares regression line relating 𝑦 = weight to 𝑥 = height is
𝑦̂ = −186.478 + 4.706𝑥
a) Predict the weight for Veronica, a woman who is 63 inches tall.
̂ = −𝟏𝟖𝟔. 𝟒𝟕𝟖 + 𝟒. 𝟕𝟎𝟔(𝟔𝟑) = 𝟏𝟏𝟎
𝒚
Veronica, who is 63 inches tall, is predicted to weigh 110 pounds.
b) Shannon has a height of 68 inches and her residual is -15. What is her actual weight?
̂ = −𝟏𝟖𝟔. 𝟒𝟕𝟖 + 𝟒. 𝟕𝟎𝟔(𝟔𝟖) = 𝟏𝟑𝟑. 𝟓𝟑
𝒚
̂ = 𝐫𝐞𝐬𝐢𝐝𝐮𝐚𝐥
𝒚−𝒚
𝒚 − 𝟏𝟑𝟑. 𝟓𝟑 = −𝟏𝟓;
𝒚 = 𝟏𝟏𝟖. 𝟓𝟑
She actually weighs 118.5 pounds.
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4) Suppose that Clayton Kershaw of the Los Angeles Dodgers 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 can be modeled by a Normal distribution.
a) About what proportion of his fastballs will travel greater than 100 mph?
𝑵(𝟗𝟒, 𝟐)
𝐧𝐨𝐫𝐦𝐚𝐥𝐜𝐝𝐟(𝐥𝐨𝐰𝐞𝐫: 𝟏𝟎𝟎, 𝐮𝐩𝐩𝐞𝐫: 𝟏𝟎𝟎𝟎𝟎, 𝝁: 𝟗𝟒, 𝝈: 𝟐) = 𝟎. 𝟎𝟎𝟏𝟑
About 0.0013 of Kershaw’s fastballs will travel greater than 100 mph.
b) About what proportion of his fastballs will travel less than 90 mph?
𝑵(𝟗𝟒, 𝟐)
𝐧𝐨𝐫𝐦𝐚𝐥𝐜𝐝𝐟(𝐥𝐨𝐰𝐞𝐫: 𝟎, 𝐮𝐩𝐩𝐞𝐫: 𝟗𝟎, 𝝁: 𝟗𝟒, 𝝈: 𝟐) = 𝟎. 𝟎𝟐𝟐𝟖
About 0.0228 of Kershaw’s fastballs will travel less than 90 mph.
c) About what proportion of his fastballs will travel between 93 and 95 mph?
𝑵(𝟗𝟒, 𝟐)
𝐧𝐨𝐫𝐦𝐚𝐥𝐜𝐝𝐟(𝐥𝐨𝐰𝐞𝐫: 𝟗𝟑, 𝐮𝐩𝐩𝐞𝐫: 𝟏𝟎𝟎𝟎𝟎, 𝝁: 𝟗𝟓, 𝝈: 𝟐) = 𝟎. 𝟑𝟖𝟐𝟗
About 0.3829 of Kershaw’s fastballs will travel between 93 and 95 mph.
d) What is the 30th percentile of Kershaw’s distribution of fastball velocities?
𝑵(𝟗𝟒, 𝟐)
𝐢𝐧𝐯𝐍𝐨𝐫𝐦(𝐚𝐫𝐞𝐚: 𝟎. 𝟑, 𝝁: 𝟗𝟒, 𝝈: 𝟐) = 𝟗𝟐. 𝟗𝟓
The 30th percentile of Kershaw’s fastballs is 92.95 mph.
e) Suppose that a different pitcher’s fastballs have a mean velocity of 92 mph and 40% of his
fastballs go less than 90 mph. What is his standard deviation of his fastball velocities, assuming
his distribution of velocities can be modeled by a Normal distribution?
𝐢𝐧𝐯𝐍𝐨𝐫𝐦(𝐚𝐫𝐞𝐚: 𝟎. 𝟒, 𝝁: 𝟎, 𝝈: 𝟏) = −𝟎. 𝟐𝟓
𝟗𝟎 − 𝟗𝟐
−𝟎. 𝟐𝟓 =
𝝈
𝝈 = 𝟖 𝐦𝐩𝐡
The standard deviation of fastball velocities is 8 mph.