What a hypothesis test is Problem set A

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.

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.

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.

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

1. (a) \(H_1: \mu > 12\)
2. (b) \(H_1: \mu \neq 12\)
3. (c) \(H_1: \mu < 12\)

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 \(p = 0.034\)
2. (b) \(\alpha = 5\%\) and \(p = 0.081\)
3. (c) \(\alpha = 1\%\) and \(p = 0.012\)

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.
2. (b) Write this as a proportion of the 500 samples.
3. (c) Is a sample mean of 63 extreme enough, at the 5% level, to doubt \(H_0\)? Explain.

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.
2. (b) Write the conclusion in context.

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 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?