Unbiased estimates, validity and reliability Problem set B

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

1. (a) (a) 'Which film is the best ever made?'
2. (b) (b) 'How many bedrooms are there in your home?'

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

1. (a) (a) 26, 31, 18, 35, 29, 22
2. (b) (b) −2, 7, −9, 4, 1, −5, 3

The masses of 9 parcels have standard deviation \(s_n=0.84\) kg, found by dividing by n. Find the unbiased estimate \(s_{n-1}\) of the population standard deviation.

A sample's unbiased standard deviation \(s_{n-1}\) is 1.0247 times its \(s_n\). Find the sample size n.

In six races a swimmer's lengths-per-minute were 21, 17, 24, 18, 26 and 20. These are a random sample of her results.

1. (a) Find \(\sum(x-\bar x)^2\) and hence the unbiased estimate of the variance of her lengths per minute.
2. (b) For a different sample of 8 races, \(\sum x=96\) and \(\sum x^2=1210\). Find the unbiased estimate of the variance.

A park ranger catches trout to estimate the mean length of all trout in a region. The lengths (cm) of eight trout are 31.2, 28.5, 35.0, 30.1, 33.4, 29.8, 32.6, 30.9.

1. (a) Find an unbiased estimate of the population mean.
2. (b) Find an unbiased estimate of the population variance.
3. (c) The ranger caught all eight trout in one pond. Explain why the estimates may not be valid.
4. (d) How could the reliability of the result in part (a) be tested?

A leisure centre leaves a questionnaire at the pool entrance: 'How many times did you swim at the centre last month?' After one week the responses were 9, 14, 6, 11, 10, 640.

1. (a) Explain why it is reasonable to remove the value 640.
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 teacher gives the students two different versions of a vocabulary test, each with 30 questions on the same topics, on the same day, 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 tutor asks at the end of a course: 'Do you think this course will improve your exam grade?'

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 tutor later compares the answers with an independent measure, the students' real exam grades. What aspect of validity is being tested by this comparison?

A calculator's 1-Variable Stats for the heights of 9 randomly chosen seedlings shows \(\sigma_x=0.620\) cm and \(s_x=0.658\) cm.

1. (a) Which of the two values is used to estimate the standard deviation of all seedlings? 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 masses (kg) of 10 suitcases are summarised by \(\sum x=130\) and \(\sum x^2=1790\). 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.