Unbiased estimates, validity and reliability Problem set B Answers

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

1. (a) Mean 26.8; variance 38.2
2. (b) Mean −0.143; variance 30.8

\(s_{n-1}=0.891\)

\(n=21\)

1. (a) \(\sum(x-\bar x)^2=60\); \(s_{n-1}^2=12\)
2. (b) \(s_{n-1}^2=8.29\)

1. (a) \(\bar x=31.4\) cm
2. (b) \(s_{n-1}^2=4.48\)
3. (c) All the trout came from one pond, so the sample may not represent trout in the whole region: the estimates may not be valid.
4. (d) Catch a second random sample from other ponds (or measure again) and check that the estimate is close to the first.

1. (a) 640 swims in a month is over 20 a day: impossible, so it is an error.
2. (b) Mean 10.0; variance 8.50
3. (c) 'Swim' is not defined (does a short dip count?) and people may not remember, so the question is not valid.
4. (d) Only people who came in, and chose to post it back, replied: not random, so the sample is biased and not valid.
5. (e) Over-estimate: regular swimmers are at the pool more and more likely to reply.
6. (f) Repeat the survey in other weeks and with more swimmers, and check that the estimates agree.

1. (a) Reliability is the consistency of results: measuring again in the same way gives similar results.
2. (b) Parallel forms
3. (c) Test-retest: give the same test again later and compare. Or internal consistency: check that questions on the same topic 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 the grade; it is a leading question, so people who have just finished tend to say yes; 'improve' 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 tutor compares the answers with an independent measure, the real grades.

1. (a) \(s_x=0.658\) (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) \(s_{n-1}^2=0.433\)

1. (a) \(\bar x=13.0\)
2. (b) \(s_{n-1}^2=11.1\)