Unbiased estimates, validity and reliability Problem set A

Say whether each survey question is valid and whether its results are reliable. Justify each answer.

1. (a) (a) 'Which is the most beautiful city in Europe?'
2. (b) (b) 'How many pets live in your home?'

Find unbiased estimates of the population mean and variance from each sample. Give each to 3 significant figures.

1. (a) (a) 14, 19, 8, 23, 17, 11
2. (b) (b) 5, −3, 8, −6, 2, 0, −4

The heights of 14 saplings have standard deviation \(s_n=6.3\) cm, found by dividing by n. Find the unbiased estimate \(s_{n-1}\) of the population standard deviation.

For a sample, \(s_{n-1}\) is 1.0328 times \(s_n\). Find the sample size n.

In six matches a team scored 14, 10, 17, 12, 9 and 16 goals. These are a random sample of its scores.

1. (a) Find \(\sum(x-\bar x)^2\) and hence the unbiased estimate of the variance of the number of goals per match.
2. (b) For a different sample of 10 matches, \(\sum x=120\) and \(\sum x^2=1500\). Find the unbiased estimate of the variance.

Karim weighs bags of flour from one supermarket to estimate the mean mass of all bags sold in his town. The masses (g) of seven bags are 1004, 998, 1010, 995, 1002, 1008, 1001.

1. (a) Find an unbiased estimate of the population mean.
2. (b) Find an unbiased estimate of the population variance.
3. (c) Karim took all seven bags from one supermarket. Explain why the estimates may not be valid.
4. (d) How could the reliability of the result in part (a) be tested?

A bus company leaves a questionnaire on its buses for passengers to post back: 'How many journeys did you make on our buses last month?' After one week the responses were 22, 15, 30, 18, 25, 1350.

1. (a) Explain why it is reasonable to remove the value 1350.
2. (b) For the remaining values, find unbiased estimates of the population mean and variance.
3. (c) Comment on the content validity of the question.
4. (d) Comment on the validity of the sampling method.
5. (e) Is the estimate of the mean more likely to be an over-estimate or an under-estimate? Justify.
6. (f) How could the reliability of the survey be improved?

A psychologist gives the same 20-question personality test to a group of students, then gives them the same test again two weeks later and compares the two sets of scores.

1. (a) Define reliability.
2. (b) Name the reliability test used.
3. (c) Suggest a different way to assess the reliability of the test.

A fitness trainer asks at the end of a programme: 'Do you think this programme will improve your fitness?'

1. (a) Define validity.
2. (b) Give two reasons why this question does not have content validity.
3. (c) Explain why making the questionnaire anonymous might improve its validity.
4. (d) The trainer later compares the answers with an independent measure of fitness, such as a fitness test. What aspect of validity is being tested by this comparison?

A calculator's 1-Variable Stats for the times of 8 randomly chosen runners shows \(\sigma_x=1.84\) s and \(s_x=1.97\) s.

1. (a) Which of the two values is used to estimate the standard deviation of all runners? Give a reason.
2. (b) Explain why the standard deviation that divides by n tends to be too small as an estimate of the population value.
3. (c) Find the unbiased estimate of the population variance.

The times (seconds) of 8 pit stops are summarised by \(\sum x=96\) and \(\sum x^2=1260\). The raw list is lost.

1. (a) Find an unbiased estimate of the population mean.
2. (b) Find an unbiased estimate of the population variance.