Question

Students in Physics and Chemistry a random sample was taken from 20 students in Physics, from which an average of 3 hours per week was obtained with a deviation. typical of 2.5 hours, and another sample of 30 independent chemistry students from the previous one, which presented an average of 2.8 hours, with a deviation. typical 2.7h. Assuming that the weekly study hours of the two types of students follow a normal distribution,

(a) Calculate a 95% confidence interval for the variance ratio of weekly study hours for physics students and chemistry students. (22pts)

(b) Assuming that the variances of the hours of study are similar, obtain a confidence interval of 95% for the difference of means of hours of study of the two types of students. (8pts)

Answer 1

a)

We need to construct the 95% confidence interval for the ratio of the two population variances. The following sample information is provided:

Sample Standard Deviation 1	2.5
Sample Size 1	20
Sample Standard Deviation 2	2.7
Sample Size 2	30

The critical values for α=0.05 and df_1 = n_1 - 1 = 20 - 1 = 19 and df_2 = n_2 - 1 = 30 - 1 = 29 degrees of freedom are:

The corresponding 95\%95% confidence interval is computed as follows:

CI = (0.3842, 2.0593)

We need to construct the 95% confidence interval for the difference between the population means μ1​−μ2​, for the case that the population standard deviations are not known. The following information has been provided about each of the samples:

Sample Mean 1	3
Sample Mean 2	2.8

Based on the information provided, we assume that the population variances are equal, so then the number of degrees of freedom are df = n_1 + n_2 -2 = 20 + 30 - 2 = 48

The critical value for α=0.05 and df = 48 degrees of freedom is t_c =​ 2.011. The corresponding confidence interval is computed as shown below:

Since the population variances are assumed to be equal, we need to compute the pooled standard deviation, as follows:

Since we assume that the population variances are equal, the standard error is computed as follows:

Now, we finally compute the confidence interval:

CI = (-1.322, 1.722)