Name: _______________________________________________ Per: ____ Date: ______________
Chapter 8 Problem Set
A.P. Statistics
1) Suppose you know that the distribution of finishing times for a certain crossword puzzle has a mean
of 25 minutes, a standard deviation of 8 minutes, and is moderately skewed left. You take an SRS of
45 finish times from this distribution and calculate the mean finish time, 𝑥̅ .
(a) Describe the shape, center, and spread of the sampling distribution of 𝑥̅ .
(b) Find a number, k, such that 95% of the values in the sampling distribution will lie within k
minutes of the mean of the distribution.
(c) If you take repeated samples of size 45 from this population, what proportion of the time will
the interval 𝑥̅ ± 𝑘 contain the number 25? Explain.
2) An insect ecologist reports a 95% confidence interval for the mean length of full-grown aquatic larvae
of the Phantom Midge Chaoborus albatusto be 6.9 to 8.5 mm, based on a sample of 9 individual larvae.
(a) What are the point estimate and margin of error associated with this confidence interval?
(b) The ecologist stated that “all necessary conditions for constructing this confidence interval were
met.” What does this tell you about his methods and about the population of insect larvae?
(c) If the ecologist had reported a 99% confidence interval instead of a 95% interval, how would it
have been different? Explain.
(d) The ecologist was unhappy with how wide this interval was. What should he do to produce a
narrower interval with the same level of confidence? Explain
3) A New York Timespoll on women’s issues interviewed 1025 women randomly selected from the
United States, excluding Alaska and Hawaii. The poll found that 52% of the women said they do not get
enough exercise.
(a) Construct and interpret a 99% confidence interval that estimates the proportion of women in the
United States who do not feel that they get enough exercise. Use the Four-step process.
(b) Explain, in the context of this problem, what “99% confidence” means.
(c) Suppose this poll was conducted by telephone calls made from 9 am to 5 pm. Explain how using
this method might result in biased results, and speculate about the direction of bias.
4) You construct three 88% confidence intervals as follows:
A) A t-interval with 6 df.
B) A t-interval with 2 df.
C) A z-interval
Assuming the mean and standard deviation are the same for all three intervals, write the three intervals
(A, B, and C) in order, from narrowest to widest.
5) You are sampling from a population with a known standard deviation of 20 and want to construct a
95% confidence interval with a margin of error of no more than 4. What is the smallest sample that
will produce such an interval?
6) Below are graphical representations of three different samples from three different populations. In
each case, discuss whether the Normality condition for constructing a t confidence interval has been
satisfied.
7) About 130,000 high school students took the AP Statistics exam in 2010. The free-response section of
the exam consisted of five open-ended problems and an investigative task. Each free-response question is
scored on a 0 to 4 scale (with 4 being the best). For one of the problems, a random sample of 30 student
papers yielded the scores thatare graphed in the dot plot of part (a) in the previous problem. The mean
score for this sample is 1.267 and the standard deviation is 1.230.
(a) Find and interpret the standard error of the mean.
(b) Construct and interpret a 99% confidence interval to estimate the mean score on this question.
8) A husband and wife, Mike and Lori, share a digital music player that has a feature that randomly
selects which song to play. A total of 2,384 songs were loaded onto the player, some by Mike and the rest
by Lori. Suppose that when the player was in the random-selection mode, 13 of the first 50 songs selected
were songs loaded by Lori.
(a) Construct and interpret a 90 percent confidence interval for the proportion of songs on the player
that were loaded by Lori.
(b) Mike and Lori are unsure about whether the player samples the songs with replacement or without
replacement when the player is in random-selection mode. Explain why this distinction is not important
for the construction of the interval in part (a).
9) A humane society wanted to estimate with 95 percent confidence the proportion of households in its
county that own at least one dog.
(a) Interpret the 95 percent confidence level in this context.
The humane society selected a random sample of households in its county and used the sample to
estimate the proportion of all households that own at least one dog. The conditions for calculating a 95
percent confidence interval for the proportion of households in this county that own at least one dog
werechecked and verified, and the resulting confidence interval was 0.417 ± 0.119
(b) A national pet products association claimed that 39 percent of all American households owned at
least one dog. Does the humane society’s interval estimate provideevidence that the proportion of
dog owners in its county is different from the claimed national proportion? Explain.
(c) How many households were selected in the humane society’s sample? Show how you obtained
your answer.