Variances and Chi-Square Tests

72. A random sample of 20 students obtained a mean of x = 72 and a variance of s2 = 16 on a college placement test in mathematics. Assuming the scores to be normally distributed, construct a 98% confidence interval for sigma^2.

78. Construct a 90% confidence interval for (sigma^2)1/(sigma^2)2 in Exercise 43. Were we justified in assuming that (sigma^2)1 does not equal (sigma^2)2 when we constructed our confidence interval for mu1 - mu2?

[See attached file for decription and answer to Exercise 43.]

83. Three cards are drawn from an ordinary deck of playing cards, with replacement, and the number Y of spades is recorded. After repeating the experiment 64 times, the following outcomes were recorded:

y 0 1 2 3
f 21 31 12 0

Test the hypothesis of 0.01 level of significance that the recorded data may be fitted by the binomial distribution b(y; 3, 2/4), y = 0, 1, 2, 3.

85. A coin is thrown until a head occurs and the number X of tosses recorded. After repeating the experiment 256 times, we obtained the following results:

x 1 2 3 4 5 6 7 8
f 136 60 34 12 9 1 3 1

Test the hypothesis at the 0.05 level of significance that the observed distribution of X may be fitted by the geometric distribution g(x; ½), x = 1, 2, 3, ....

90. In an experiment to study the dependence of hypertension on smoking habits, the following data were taken on 180 individuals:

Non-Smokers Moderate Smokers Heavy Smokers
Hypertension 21 36 30
No Hypertension 48 26 19

Test the hypothesis that the presence or absence of hypertension is independent of smoking habits. Use a 0.05 level of significance.

92. A random sample of 200 married men, all retired, were classified according to education and number of children:

NUMBER OF CHILDREN
EDUCATION 0-1 2-3 Over 3
Elementary 14 37 32
Secondary 19 42 17
College 12 17 10

Test the hypothesis, at the 0.05 level of significance, that the size of a family is independent of the level of education attained by the father.

Please see attached file for full problem description.