Statistics I-HW III
Due 2025.08.05 18:00
1. (30%) According to the regulations of the Executive Yuan Environmental
Protection Agency, the average concentration of lead in drinking water should not
exceed 10 units. Now, 30 observations are collected. The sample mean x 9.60
and sample standard deviation s 1.00 units are obtained.
(a) Obtain a 99% confidence interval for the population mean. (5%)
(b) If the width of confidence interval in (a) is too wide, what can we do to reduce
the interval width? (5%)
(c) Under 0.01 , can it be said that drinking water meets the requirements?
Perform hypothesis testing. (5%)
(d) Why the value of considered in (c) is very low? (5%)
(e) What is the approximate value of p-value for x 9.60 ? (5%)
(f) Do the results by (a), (c) and (e) are consistent? (5%)
2. (30%) To learn about the feeding habit of bars, 72 bats were tagged and tracked by
radio. 36 female and 36 male bats are selected. The distances flown (in meters)
between feedings were noted for each of the 72 bats, and the follow summary
statistics were obtained.
Female Bats: n=36, x1 180 , s1 92
Male Bats: n=36, x2 136 , s2 86
Test the hypothesis that the mean distance flown between feeding is the same for
the populations of both male and of female bats ( 0.05 ).
(a) State the null and alternative hypotheses. (5%)
(b) What test statistic would you to test the hypothesis? (5%)
(c) What is the sampling distribution of the considered statistic in (b)? (5%)
(d) Find the p-value (5%)
(e) What is the critical region? (5%)
(f) Do the results by (d) and (e) are consistent? (5%)
3. (10%) A quality engineer in a light bulb manufacturing factory plans a study to
estimate the average life of a large shipment of light bulbs. The engineer wants to
estimate the average life within plus or minus 15 hours with 95% confidence. If
the process standard deviation is 100 hours, please determine the minimum sample
size for this study. If the confidence is 90%, what the reduced sample size is?
4. (30%) Given a coin, we want to test whether it is fair. Let p denotes the
probability of “head event”. It is tossed 10 times. Suppose the decision rule is
“Say it is significantly not fair if the number of “head event”
is less than 2 or greater than 8.”
(a) What are the null and alternative hypotheses? (5%)
(b) What is the test statistic? (5%)
(c) What is the distribution of the test statistic? (5%)
(d) Compute the probability of type I error. (5%)
(e) Compute the probability of type II error as p 0.8 . (5%)
(f) If the number of “head event” you observed is 0, what is the p -value? (5%)
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