What a hypothesis test is Problem set B Worked solutions

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.

• The claim: \(H_0: \mu = 40\)
• Not as long means lower: \(H_1: \mu < 40\)
• One-tailed (lower tail)

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 survey value: \(H_0: \mu = 3.5\)
• Higher: \(H_1: \mu > 3.5\)
• One-tailed (upper tail)

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.

• Old mean: \(H_0: \mu = 72\)
• Changed, either way: \(H_1: \mu \neq 72\)
• Two-tailed

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

1. (a) \(H_1: \mu > 45\)

   • \(>\) points up: upper tail

2. (b) \(H_1: \mu \neq 45\)

   • \(\neq\) allows both directions: two-tailed

3. (c) \(H_1: \mu < 45\)

   • \(<\) points down: lower tail

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 \(p = 0.027\)

   • \(0.027 < 0.05\), so reject \(H_0\)

2. (b) \(\alpha = 5\%\) and \(p = 0.064\)

   • \(0.064 > 0.05\), so do not reject \(H_0\)

3. (c) \(\alpha = 0.5\%\) and \(p = 0.008\)

   • The level is 0.5% = 0.005. \(0.008 > 0.005\), so do not reject \(H_0\)

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.

   • Bars from 174 upwards: 7 + 2 = 9

2. (b) Write this as a proportion of the 400 samples.

   • 9 ÷ 400 = 0.0225

3. (c) Is a sample mean of 174 cm extreme enough, at the 5% level, to doubt the school's claim? Explain.

   • 0.0225 is smaller than 0.05.
   • A result this extreme happens rarely if the claim is true: there is evidence the mean is higher than 170 cm

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.

   • \(H_0: \mu = 8\)
   • Longer means higher: \(H_1: \mu > 8\)
   • One-tailed (upper)

2. (b) Write the conclusion in context.

   • \(0.022 < 0.05\), so reject \(H_0\)
   • There is sufficient evidence, at the 5% level, that the mean advert time is longer than 8 minutes

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?

• Equal numbers of each colour give 40 of each in 160 draws
• The counts 43, 37, 41, 39 are all close to 40
• Not extreme: consistent with equal numbers

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?

• Equal numbers of each colour give 40 of each in 160 draws
• 14 is 26 below 40 and 66 is 26 above it
• Counts this far away would be very unlikely: not consistent with equal numbers