What a hypothesis test is Problem set A Worked solutions

A cereal box is labelled with a mean mass of 500 g. An inspector thinks the boxes are underfilled. Write \(H_0\) and \(H_1\) for the mean mass \(\mu\) and say whether the test is one-tailed or two-tailed.

• The claim is 500 g: \(H_0: \mu = 500\)
• Underfilled means lower: \(H_1: \mu < 500\)
• One direction only: one-tailed (lower tail)

A bus company says the mean journey time on a route is 32 minutes. Passengers think it is now longer. Write \(H_0\) and \(H_1\) for the mean time \(\mu\) and say whether the test is one-tailed or two-tailed.

• The company's claim: \(H_0: \mu = 32\)
• Longer means higher: \(H_1: \mu > 32\)
• One-tailed (upper tail)

A factory makes bolts with a mean length of 6.0 cm. After a machine is reset the manager wants to know whether the mean length has changed. Write \(H_0\) and \(H_1\) and say whether the test is one-tailed or two-tailed.

• Old mean: \(H_0: \mu = 6.0\)
• Changed, either way: \(H_1: \mu \neq 6.0\)
• Two-tailed

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

1. (a) \(H_1: \mu > 12\)

   • \(>\) points up: upper tail

2. (b) \(H_1: \mu \neq 12\)

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

3. (c) \(H_1: \mu < 12\)

   • \(<\) 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 \(p = 0.034\)

   • \(0.034 < 0.05\), so reject \(H_0\)

2. (b) \(\alpha = 5\%\) and \(p = 0.081\)

   • \(0.081 > 0.05\), so do not reject \(H_0\)

3. (c) \(\alpha = 1\%\) and \(p = 0.012\)

   • The level is 1% = 0.01. \(0.012 > 0.01\), so do not reject \(H_0\)

A test is designed so that the mean score is 60 when \(H_0\) is true. A computer simulates 500 random samples of 25 students under \(H_0\) and records each sample mean. The histogram shows the 500 sample means. A real sample has mean score 63.

1. (a) Add up the frequencies for sample means of 63 or more.

   • Bars from 63 upwards: 9 + 2 + 0 = 11

2. (b) Write this as a proportion of the 500 samples.

   • 11 ÷ 500 = 0.022

3. (c) Is a sample mean of 63 extreme enough, at the 5% level, to doubt \(H_0\)? Explain.

   • 0.022 is smaller than 0.05.
   • A result this extreme happens rarely under \(H_0\): there is evidence the mean is higher than 60

A teacher says the mean mark in her maths test is 65. A colleague thinks a revision programme has raised the mean. A test gives \(p = 0.018\) at the 5% significance level.

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

   • \(H_0: \mu = 65\)
   • Raised means higher: \(H_1: \mu > 65\)
   • One-tailed (upper)

2. (b) Write the conclusion in context.

   • \(0.018 < 0.05\), so reject \(H_0\)
   • There is sufficient evidence, at the 5% level, that the mean mark is higher than 65

A spinner has four equal sectors numbered 1 to 4. It is spun 200 times. The bar chart shows the results and the dashed line shows 50, the frequency for each number if the spinner is fair. Do the results look consistent with a fair spinner?

• A fair spinner gives 50 of each number
• The counts 48, 52, 46, 54 are all close to 50
• These results are not extreme: consistent with a fair spinner

A different 4-sector spinner is spun 200 times. The bar chart shows the results and the dashed line shows 50. Do these results look consistent with a fair spinner?

• A fair spinner gives 50 of each number
• 19 is 31 below 50 and 83 is 33 above it
• Counts this far away would be very unlikely: not consistent with a fair spinner