Unbiased estimates, validity and reliability Problem set A Answers

1. (a) Not valid and not reliable
2. (b) Valid and reliable

1. (a) Mean 15.3; variance 29.9
2. (b) Mean 0.286; variance 25.6

\(s_{n-1}=6.54\)

\(n=16\)

1. (a) \(\sum(x-\bar x)^2=52\); \(s_{n-1}^2=10.4\)
2. (b) \(s_{n-1}^2=6.67\)

1. (a) \(\bar x=1003\) g
2. (b) \(s_{n-1}^2=28.0\)
3. (c) Karim used only one supermarket, so the sample may not represent all the bags sold in the town: the estimates may not be valid.
4. (d) Take a second random sample (or repeat the weighing) and check that the estimate is close to the first.

1. (a) 1350 journeys in a month is over 40 a day: impossible, so it is an error.
2. (b) Mean 22.0; variance 34.5
3. (c) 'Journey' is not defined (is a return trip one or two?) and people may not remember exactly, so the question is not valid.
4. (d) Only bus passengers who chose to post it back replied: not random, so the sample is biased and not valid.
5. (e) Over-estimate: people who travel often are on the buses more and more likely to reply.
6. (f) Repeat the survey in other weeks and with more passengers, and check that the estimates agree.

1. (a) Reliability is the consistency of results: repeating the test in the same conditions gives similar results.
2. (b) Test-retest
3. (c) Parallel forms: give two different versions of the test and compare. Or internal consistency: check that questions on the same trait agree with each other.

1. (a) Validity is how well a test or question measures what it is meant to measure.
2. (b) It asks for an opinion about the future, not a measure of fitness; it is a leading question, so people who have just finished tend to say yes; 'fitness' is not defined.
3. (c) People are more likely to answer honestly if no one can trace the answer to them, so the answers are closer to what they really think.
4. (d) Criterion-related validity: the trainer compares the answers with an independent measure of fitness.

1. (a) \(s_x=1.97\) (this is \(s_{n-1}\)): it divides by \(n-1\), so it estimates the population value.
2. (b) The data lie closer to the sample mean than to the true mean, so the squared deviations add to too little. Dividing by n does not correct for this.
3. (c) \(s_{n-1}^2=3.88\)

1. (a) \(\bar x=12.0\)
2. (b) \(s_{n-1}^2=15.4\)