What a hypothesis test is Problem set B

A battery maker claims its batteries last a mean of 40 hours. A reviewer thinks they do not last as long. Write \(H_0\) and \(H_1\) for the mean life \(\mu\) and say whether the test is one-tailed or two-tailed.

A survey found that teenagers spend a mean of 3.5 hours a day on their phones. A researcher thinks the mean is now higher. Write \(H_0\) and \(H_1\) for the mean time \(\mu\) and say whether the test is one-tailed or two-tailed.

The mean resting heart rate in a town is 72 beats per minute. After a gym opens a doctor wants to know whether the mean has changed. Write \(H_0\) and \(H_1\) and say whether the test is one-tailed or two-tailed.

For each alternative hypothesis, say whether the test is one-tailed (upper), one-tailed (lower) or two-tailed.

1. (a) \(H_1: \mu > 45\)
2. (b) \(H_1: \mu \neq 45\)
3. (c) \(H_1: \mu < 45\)

In each case a hypothesis test gives a p-value. Say whether you reject \(H_0\) and write what that means for \(H_1\).

1. (a) \(\alpha = 5\%\) and \(p = 0.027\)
2. (b) \(\alpha = 5\%\) and \(p = 0.064\)
3. (c) \(\alpha = 0.5\%\) and \(p = 0.008\)

A school says the mean height of its students is 170 cm. A computer simulates 400 random samples of 30 students under this claim and records each sample mean. The histogram shows the 400 sample means. A real sample has mean height 174 cm.

1. (a) Add up the frequencies for sample means of 174 or more.
2. (b) Write this as a proportion of the 400 samples.
3. (c) Is a sample mean of 174 cm extreme enough, at the 5% level, to doubt the school's claim? Explain.

A cinema says the mean time for its adverts is 8 minutes. A customer thinks the adverts now run longer. A test gives \(p = 0.022\) at the 5% significance level.

1. (a) Write \(H_0\) and \(H_1\) and say whether the test is one-tailed or two-tailed.
2. (b) Write the conclusion in context.

A bag holds equal numbers of red, blue, green and yellow counters. A counter is drawn, noted and replaced, 160 times. The bar chart shows the results (1 red, 2 blue, 3 green, 4 yellow) and the dashed line shows 40. Do the results look consistent with equal numbers of each colour?

A second bag is sampled the same way, 160 draws with replacement. The bar chart shows the results (1 red, 2 blue, 3 green, 4 yellow) and the dashed line shows 40. Do these results look consistent with equal numbers of each colour?