Estimating the ratio of two variances

Question

Assignment:

Q1. One-and Two-Sample Estimation Problems: Two Samples: Estimating the Ratio of Two Variances

Q2. 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 o².

Q3. Construct a 90% confidence interval for o²₁ / o²₂ in

Q: Were we justified in assuming that o²₁ does not equal o²₂ when we constructed our confidence interval for μ₁ - μ₂? A taxi company is trying to decide whether to purchase brand A or brand B tires for its fleet of taxis. To estimate the difference in the two brands, an experiment is conducted using 12 of each brand. The tires are run until they wear out. The results are

Brand A: x1 = 36,300 kilometers,

S1 = 5,000 kilometers.

Brand B: x2 = 38, 100 kilometers

S2 = 6, 100 kilometers.

Compute a 95% confidence interval for μA - μB assuming the populations to be approximately normally distributed. You may not assume that the variances are equal.

Q4.  Three cards are drawn from an ordinary deck of playing cards, whith 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.

Q5. 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, ....

Q6. 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.

Q7. 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.