Psychology 406 Assignment #6

1) Hays — Chapter 6, #22; Chapter 7, #26

2) Suppose you develop a test of spatial reasoning and administer it to all second grade children in the Champaign/Urbana public schools. Assume that you obtain N = 625 subjects and calculate both a sample mean of 90 and an unbiased sample variance of 16, i.e., M = 90, s^{2} = σ̂^{2} = 16. There are enough subjects to use our large sample result concerning s^{2} as an estimate of σ^{2}.

a) Under the assumptions of normality for the underlying test scores, what are 95% and 99% confidence intervals for the mean μ? What could you say if we could not assume normality of the test scores?

b) With how many σ units of the true mean will the sample estimate fall with probability .95 (Hays, pp. 256–257)

c) Using the 95% (99%) confidence interval in (a), what simple “null” hypotheses regarding μ could you reject at .05 (.01)?

d) To what group of children do your results generalize? (Hint: What type of sampling did we perform?)

3) Subjective ratings on a 1 to 10 scale given to the “goodness” of the simple figure (given below) by 34 people have a frequency table that is also given below:

[figure]

Rating Number of People
1  1
2  4
3  2
4  11
5  5
6  4
7  4
8  3
9  0
10  0

N = 34

Under the assumption that the mean rating has a normal distribution and we can estimate σ^{2} precisely enough to use the large sample estimate of s^{2}, what is a 95% confidence interval for the “true” mean μ? Would you reject the hypothesis that the figure is given the average rating of 5.5? Provide the p-value.

4) For the two figures given below, we have the responses of 34 individuals.

[figure]

Person  Person
1  4 5  18  4 10
2  4 7  19  7 8
3  4 5  20  7 8
4  4 4  21  5 8
5  6 8  22  8 7
6  3 5  23  4 5
7  5 9  24  8 9
8  4 7  25  6 9
9  7 10 26  2 8
10 4 9  27  8 10
11 6 7  28  4 9
12 7 10 29  2 7
13 5 7  30  7 9
14 2 9  31  3 5
15 6 9  32  7 5
16 8 1  33  3 6
17 4 7  34  5 8
Total 173 250

a) Perform a paired t-test of the difference between the mean ratings at α = .05. Construct the corresponding 95% confidence interval for the true mean difference.

b) Assume (incorrectly, of course) that we actually obtained these ratings from two independent samples of subjects. Perform the appropriate t-test of the difference between mean ratings at α = .05. Construct the corresponding 95% confidence interval for the true mean difference. Are the standard deviations in the two groups comparable? How do the results based on this test compare with the paired t-test in (a)? Can you explain any difference? Include a discussion of the reduction in the degrees of freedom and the differing denominators for the two test statistics.