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AP Statistics
7.1B Assignment
1. During World War II, 12,000 able-bodied male undergraduates at the University of Illinois participated in
required physical training. Each student ran a timed mile. Their times followed the Normal distribution
with mean 7.11 minutes and standard deviation 0.74 minutes. An SRS of 100 of these students has mean
time x  7.15 minutes. A second SRS of size 100 has mean x  6.97 minutes. After many SRSs, the
values of the sample mean x follow the Normal distribution with mean 7.11 minutes and standard
deviation 0.074 minutes.
A. What is the population? Describe the population distribution.
The population is the 12,000 able-bodied male undergraduates at the University of Illinois who
participated in the required physical training.
The population distribution is normal with mean 7.11 minutes and standard deviation 0.74 minutes
and describes the time it takes randomly selected individuals to run a mile.
B. Describe the sampling distribution of x . How is it different from the population distribution?
The sampling distribution is approximately Normal with mean 7.11 minutes and standard deviation
0.074 minutes. The mean is the same but the standard deviation is smaller.
The sampling distribution describes the average mile-time for 100 randomly selected students. (It’s
the average of 100 runners instead of individual runners.)
2. A statistics teacher fills a large container with 1000 white and 3000 red beads and then mixes the beads
thoroughly. She then has her students take repeated SRSs of 50 beads from the container. After many
SRSs, the value of the sample proportion p̂ of the red beads are approximated well by a Normal
distribution with mean 0.75 and standard deviation 0.06.
A. What is the population? Describe the population distribution.
The population is the 4000 beads in the container and the distribution is 1000 white and 3000 red
beads.
B. Describe the sampling distribution of p̂ . How is it different from the population distribution?
The sampling distribution of p̂ is approximately normal with mean 0.75 and standard deviation 0.06.
The mean is the same as the population proportion. The sample proportion is a numerical variable so
its distribution could be shown using a histogram or dotplot. The color of the individual beads is a
categorical variable and so its distribution would be best shown with a bar graph.
3. The Internal Revenue Service plans to examine an SRS of individual federal income tax returns from each
state. One variable of interest is the proportion of returns claiming itemized deductions. The total
number of tax returns in each state varies from over 15 million in California to about 240,000 in Wyoming
A. Will the sampling variability of the sample proportion change from state to state if an SRS of 2000 tax
returns is selected in each state? Explain your answer.
The sampling variability will be the same because the sample size is the same.
B. Will the sampling variability of the sample proportion change from state to state if an SRS of 1% of all
tax returns is selected in each state? Explain your answer.
The sampling variability will be different (smaller for states with a larger number of returns) because
the sample size will be different.
4. A study of the health of teenagers plans to measure the blood cholesterol levels of an SRS of 13- to 16year-olds. The researchers will report the mean x from their sample as an estimate of the mean
cholesterol level  in this population.
A. Explain to someone who knows no statistics what it means to say that x is an unbiased estimator of  .
If we choose many samples, the average of the sample means from these samples will be close to the
average of all teenagers.
B. The sample result x is an unbiased estimator of the population mean  no matter what size SRS the
study chooses. Explain to someone who knows no statistics why a large random sample gives more
trustworthy results than a small random sample.
A larger sample will give more information and, therefore, be more precise. The variability in the
distribution of the sample average decreases as the sample size increases.