Question

Surveys of students’ hours of studying has the following results: a sample of 21 econ students has a mean=25 and a variance =36; a sample of 16 poly sci students has a mean=7 and a variance=85. Are the variance of study hours the same for econ and poly sci students? Use a significance level of .01. The null hypothesis is The alternative hypothesis is the critical value of the test statistics is the value of the test statistic is your conclusion is

Answer 1

The question examines whether there is a significant difference in the variances of study hours for econ and poly sci students using variance analysis and hypothesis testing at a .01 significance level.

Step-by-step explanation:

The question primarily deals with the concepts of hypothesis testing and variance analysis in statistics. A student is trying to determine if the variances in study hours for economics and political science students are the same, using a significance level of .01. They are applying a statistical test of variance, such as the F-test, to compare the variances of the two independent samples.

To form the null hypothesis (H0), the student would posit that there is no difference in the variances of study hours between the two groups. The alternative hypothesis (H1) suggests that there is a difference. The critical value is the cut-off value from the F-distribution table at the 0.01 level of significance which determines whether the observed test statistic falls into the rejection area. The test statistic is calculated using the variances provided and the respective sample sizes.

The test statistic value and whether it exceeds the critical value will lead to the conclusion of either rejecting or not rejecting the null hypothesis. If the value of the test statistic is greater than the critical value, the null hypothesis would be rejected, indicating a significant difference in variances. If not, the null hypothesis would not be rejected, indicating no significant difference in variances.

Answer 2

Step-by-step explanation:

Hello!

The objective of these surveys is to compare if the population variances of the studying hours of econ students and poly sci students are equal.

To do this analysis you have to conduct a variance ratio test.

Be:

X₁: Time in hours that an economy students studies

n₁: 21 students

X[bar]₁= 25 hs

S₁²= 36 hs

And

X₂: Time in hours that a poly sciences students studies

n₂: 16 students

X[bar]₂= 7 hs

S₂²= 85 hs

The hypotheses are:

H₀: σ₁² = σ₂²

H₁: σ₁² ≠ σ₂²

α: 0.01

The test statistic is a Snedecor's F

`F= ((S^2_1)/(S^2_2)) *((Sigma^2_1)/(Sigma^2_2)) ~~ F_(n_1-1;n_2-1)`

This test is two-tailed so the rejection region is divided in two tails:

`F_(n_1-1;n_2-1:\alpha /2)= F_(20;15;0.005)= 0.29`

`F_(n_1-1;n_2-1;1-\alpha /2)= F_(20;15;0.995)= 3.88`

So, you will reject the null hypothesis if
`F_(H_0)` ≤ 0.29 or if
`F_(H_0)` ≥ 3.88

And you will not reject the null hypothesis if 0.29 <
`F_(H_0)` < 3.88

`F_(H_0)= ((36)/(85) )*1= 0.42`

This value is between 0.29 and 3.88, then the decision is to not reject the null hypothesis.

Using a 1% level of significance, there is no significant evidence to reject the null hypothesis. You can conclude that the population variances of the time the economy and poly science students spend studying are the same.

I hope this helps!