Statistics 104, Homework 2
Due : January 24th, 2018, In Class
This homework is to practice on the binomial test. Total: 90 points.
1. (40 points) Understanding the binomial test. Every weekday, you take a bus to go to school.
The bus line comes in every 10 minutes. However, due to all kinds of reasons, the bus does
not alway arrive on time. You recorded the waiting time for 30 days and would like to test the
following hypotheses. For each hypothesis test, write down the null hypothesis and alternative
hypothesis, state the test statistic you will use, and null distribution of the test statistic, and
the rejection region at 0.05 signi cance level.
(a) Test whether half of your waiting time is at most 5 minutes.
(b) Test whether half of your waiting time is at least 5 minutes.
(c) Test whether your median waiting time is 5 minutes.
(d) Test whether 80% of your waiting time is at most 7 minutes.
(e) Test whether a quarter of your waiting time is at least 3 minutes.
2. (20 points) Data analysis. The data you collected on 30 days are as follows:
2.77, 2.32, 4.97, 0.51, 4.22, 4.35, 7.31, 3.33, 4.92, 1.53, 5.62, 7.94, 2.52, 3.59, 6.86, 6.02, 1.84,
3.22, 3.24, 6.21, 4.82, 6.4, 4.85, 6.74, 3.78, 1.54, 6.93, 7.94, 4.94, 2.5.
For each of the hypothesis in Problem 1, use the above dataset to compute the test statistic,
decide whether it is in the rejection region and draw the conclusion. In addition, for each
hypothesis, compute the p-value and compare with the signi cance level (0.05) and draw the
conclusion. Does the conclusion consistent with what you got from the rejection region.
3. (30 points) Probability calculation. Suppose you have 9 distinct observations. Let Y(1) be the
smallest observation, Y(2) be the second smallest observation, and so on:
Y(1) Compute the following three probabilities exactly.
(a) P(Y(3) < 0:5 (b) P(Y(1) < 0:5 (c) P(Y(1) < 0:3 1