Test the hypothesis H0: ^2=^2 H1: ^2>^2 using the following data. a. sx^2=125, ny=45

step 1 Here the sample is drawn from the normal population with variance sigma x square and sigma y squ

Test the hypothesis H0: ^2=^2 H1: ^2>^2 using...

Test the hypothesis $$ \begin{aligned} & H_0: \sigma_x^2=\sigma_y^2 \\ & H_1: \sigma_x^2>\sigma_y^2 \end{aligned} $$ using the following data. a. $s_x^2=125, n_y=45 ; s_y^2=51, n_y=41$ b. $s_x^2=125, n_y=45 ; s_y^2=235, n_y=44$ c. $s_x^2=134, n_y=48 ; s_y^2=51, n_y=41$ d. $\mathrm{s}_x^2=88, n_y=39 ; \mathrm{s}_y^2=167, n_y=25$

Test the hypothesis
$$
\begin{aligned}
& H_0: \sigma_x^2=\sigma_y^2 \\
& H_1: \sigma_x^2>\sigma_y^2
\end{aligned}
$$
using the following data.
a. $s_x^2=125, n_y=45 ; s_y^2=51, n_y=41$
b. $s_x^2=125, n_y=45 ; s_y^2=235, n_y=44$
c. $s_x^2=134, n_y=48 ; s_y^2=51, n_y=41$
d. $\mathrm{s}_x^2=88, n_y=39 ; \mathrm{s}_y^2=167, n_y=25$

Key Concepts

Degrees of Freedom

Degrees of freedom for each sample are determined by subtracting one from the sample size, i.e., df = n - 1. In the F-test, one uses two sets of degrees of freedom corresponding to the numerator and denominator variances. These values are critical in referring to the appropriate F-distribution to assess the significance of the computed statistic.

One-Tailed Hypothesis Testing

When testing a hypothesis where the alternative specifies a direction (e.g., one variance being greater than the other), a one-tailed test is performed. This involves comparing the calculated F-statistic to a critical value from the F-distribution, which is derived based on the chosen significance level and the respective degrees of freedom.

F-test for Equality of Variances

This is a statistical test used to compare the variances of two populations by analyzing the ratio of their sample variances. Under the null hypothesis, it is assumed that the population variances are equal. The F-test is especially useful when determining if two independent samples come from populations with equal variability.

F-statistic

The F-statistic is calculated as the ratio of two sample variances. In the context of testing for equal variances, it is typically defined as the larger sample variance divided by the smaller one to obtain a value greater than or equal to one. This statistic follows an F-distribution under the null hypothesis.

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