Question

A psychologist studied the alcohol consumption patterns of people in two age groups. One group consisted of people aged 21 to 35, and the other consisted of

people aged 36 to 50. The psychologist interviewed random and independent samples from each group. She assigned a score from 0 to 100 to each individual (a
score of 0 meant no alcohol consumption) according to factors such as the frequency and the amount of alcohol consumed. The results from the study are
summarized below.
Age 21 to 35₁-22₁-49.9 -190.4
Age 36 to 50 ₂ -21₂- = 48.8=176.9
(The first row gives the sample sizes, the second row gives the sample means, and the third row gives the sample variances.)
Assume that the scores of all people aged 21 to 35 are approximately normally distributed. Assume the same for the scores of all people aged 36 to 50.
01
the ratio of the variance of the scores of all people aged 21 to 35 to the variance of the scores of all people aged
0₂
36 to 50. Then find the lower limit and upper limit of the 99% confidence interval.
Carry your intermediate computations to at least three decimal places. Write your final responses to at least two decimal places. (If necessary, consult a list of
formulas.)
Construct a 99% confidence interval for
Lower limit:
Upper limit:
X

Answer 1

The 99% confidence interval for the ratio of the variances is:

Lower limit: 65.85 (rounded to two decimal places)

Upper limit: 71.42 (rounded to two decimal places)

Constructing the Confidence Interval

Step 1: Find the F-statistic

The F-statistic is calculated as the ratio of the two variances

F = s₁² / s₂² = 190.4 / 176.9 ≈ 1.078

Step 2: Find degrees of freedom

Degrees of freedom for the numerator: n₁ - 1 = 22 - 1 = 21

Degrees of freedom for the denominator: n₂ - 1 = 49 - 1 = 48

Step 3: Find the critical values

We need to find the two critical values of the F-distribution that define the 99% confidence interval. This requires using an F-distribution table or calculator.

For an alpha level of 0.01 (1 - 0.99) and degrees of freedom 21 and 48, the critical values are:

Lower critical value: F_L = 1.70

Upper critical value: F_U = 2.77

Step 4: Calculate the confidence limits

The confidence limits can be calculated using the following formulas:

Lower limit: s₁² / F_U * s₂²/n₂ = 190.4 / 2.77 * 176.9 / 48 ≈ 65.85

Upper limit: s₁² * F_L / n₂ = 190.4 * 1.70 / 48 ≈ 71.42

Step 5: Round the results

Therefore, the 99% confidence interval for the ratio of the variances is:

Lower limit: 65.85 (rounded to two decimal places)

Upper limit: 71.42 (rounded to two decimal places)