Math 3339Homework 5 (6.4-6.6; Chapter 7)
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• Print out this file and complete the problems.
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• Submit this assignment at http://www.casa.uh.edu under “Assignments” and choose hw5.
• Total possible points: 15
• You can use RStudio for any of these problems unless otherwise indicated.
1. The inside diameter of a randomly selected piston ring is a random variable following normal
distribution with mean value 12 cm and standard deviation 0.04 cm.
(a) If 𝑋̅ is the sample mean diameter for a random sample of n = 16 rings, where is the sampling
distribution of 𝑋̅ centered, and what is the standard deviation of the 𝑋̅ distribution?
(b) Answer the questions posted in part (a) for a sample size of n =64 rings.
(c) For which of the two random samples, the one of part (a) or the one of part (b), is 𝑋̅ more likely to
be within 0.01 cm of 12 cm? Explain your reasoning.
2. A sample of 58 data observations has a sample mean of 0.0328. If an experimenter wishes to use a
“known” value σ=0.015 for the population standard deviation, construct an appropriate 95%
confidence interval for the population mean µ.
3. The breaking strengths of a random sample of 20 bundles of wool fibers have a sample mean 436.5
and a sample standard deviation 11.9.
(a) Construct 90%, 95%, and 99% confidence intervals for the average breaking strength of the wool
fibers.
(b) Compare the widths of the three confidence intervals. At which level of confidence do you have
the widest interval?
(c) Do you think it is plausible that the average breaking strength is equal to 450?
4. An experimenter would like to construct a 99% confidence interval with a width at most 0.5 for the
average resistance of a segment of copper cable of a certain length. If the experimenter knows that the
standard deviation of such resistances is 1.55. How big a sample should the experimenter take from
the population? What happens if the standard deviation and the width of the confidence interval are
both doubled?
5. Consider the following data set which is the time (in minutes) for students to finish a math test.
34 45 27 33 38 41 45 29 30 39 34 40 28 33 36 50
(a) What is the sample mean?
(b) What is the sample standard deviation?
(c) Construct a 99% confidence interval for the population mean.
6. The “Boston” data set is a built-in dataset in the “MASS” library of R. Assume the variable “medv” in
the Boston data set is approximately normally distributed. Treating the data as a sample from a larger
normal population, find 95% confidence intervals for the variance of “medv”.
7. A sample of size 50 from a normal distribution has sample mean 5 and sample variance 20. Find 96%
confidence intervals for the variance and standard deviation of the distribution.
8. Suppose that 47% of Americans are iPhone users.
(a) If a random sample of 100 Americans are surveyed, what is the probability that the proportion of
the sample who are iPhone users is between 45% and 50%?
(b) If a random sample of 50 Americans are surveyed, what is the probability that more than 50% of
Americans are iPhone users?
9. Suppose that a random sample of 50 bottles of a particular brand of cough syrup is selected and the
alcohol content of each bottle is determined. Let µ denote the average alcohol content for the
population of all bottles of the brand under study. Suppose that the resulting 95% confidence interval
is (7.8, 9.4).
(a) Would a 99% confidence interval calculated from this same sample have been narrower or wider
than the given interval? Explain your reasoning.
(b) Consider the following statement: There is a 95% chance that µ is between 7.8 and 9.4. Is this
statement correct? Why or why not?
(c) Consider the following statement: If the process of selecting a sample of size 50 and then
computing the corresponding 95% interval is repeated 100 times, 95 of the resulting intervals will
include µ. Is this statement correct? Why or why not?
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