Statistics: HW 7
Deadline: April 18 at 8 p.m.
Problem 1. Suppose we have n data points drawn from a N(μ, 5
2
) distribution, where
the value of μ is unknown.
(a) (1 point) Suppose we have 16 data points and that the sample mean is ˉx = 20.
Construct a precise 95% confidence interval for the μ.
(b) (1 point) Explain why the confidence interval for the mean when n = 100 is always
narrower than the confidence interval when n = 10.
(c) (1 point) What is the smallest value of n so that the 95% confidence interval for the
mean will have width less than 1.0? (Still with σ2 = 52)
Problem 2. Adult mayflies live anywhere from 30 minutes to 1 day, depending on the
species. Data for one species was collected by tracking 10 mayflies. The recorded lifespans
in hours were
17.68, 13.69, 11.22, 11.05, 13.86, 14.47, 14.50, 13.47, 10.04, 13.10
(a) (1 point) Compute a 95% confidence interval for the mean lifetime of this species of
mayfly.
(b) (1 point) What assumptions did you make in part (a)?
(c) (1 point) Compute a 95% confidence interval for the standard deviation of distribution
of the lifetime of a mayfly.
(d) (1 point) Based on the sample variance of this data estimate the number of data
points you would need to make the width of the 95% confidence interval for the
mean less than or equal to 1 hour.
(e) (1 point) Is the value of n in part (d) guaranteed to be sufficient? Explain your
reasoning.
Problem 3. Finish the t-test task from the practice:
(a) (4 points) Build the distribution of p-values when H0 is true
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(b) (8 points) Build the distribution of p-values when H0 is false. Try to decrease the
sample size, while holding the mean the same. Try to increase the mean while
holding the sample size the same. Explain your results.
(c) (4 points) Build the distribution of p-values when xi ～ Exp(1) and sample size
n = 5. Explain your results.
Problem 4. Consider the z-test (slides 35–37):
(a) (4 points) Build the distribution of p-values when H0 is true. Please consider different
distributions.
(b) (8 points) Build the distribution of p-values when H0 is false. Please consider different
distributions. Try to decrease the sample size, while holding the mean the
same. Try to increase the mean while holding the sample size the same. Explain
your results.
(c) (4 points) Build the distribution of p-values when xi ～ Cauchy and sample size
n = 1000. Explain your results.
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