Question

The figure shows an \(F\) probability density function. The two dotted lines represent critical values corresponding to a two-tailed \(F\)-test at a level of significance of 0.05. The observed \(F\)-statistic for two samples is indicated by the solid line.

• A. The null hypothesis cannot be rejected.
• B. The null hypothesis is true.
• C. The ratio of the variances of the two samples is not statistically significantly different from 1.
• D. The ratio of the skewness of the two samples is not statistically significantly different from 1.

Hint:

In an \(F\)-test, the null hypothesis is rejected if the observed \(F\)-statistic exceeds the critical value. If it lies within the critical range, we cannot reject the null hypothesis.

Answer 1

The Correct Option is A, C

Step 1: Understanding the figure.
The figure shows an \(F\)-distribution with the observed \(F\)-statistic marked by the solid line. The two dotted vertical lines represent the critical values corresponding to a significance level of 0.05. This indicates a two-tailed \(F\)-test where we fail to reject the null hypothesis if the observed \(F\)-statistic falls between these two critical values.

Step 2: Interpretation of the options.
(A) The observed \(F\)-statistic lies within the range defined by the two critical values, meaning the null hypothesis cannot be rejected. This is the correct inference. (B) The null hypothesis being true refers to the assumption that there is no significant difference in variances. Since the observed \(F\)-statistic does not exceed the critical values, this is not the case. (C) This option suggests the ratio of variances is not statistically significantly different from 1, which is correct because the observed \(F\)-statistic falls within the critical range. (D) Skewness does not directly affect the interpretation of the \(F\)-test, so the ratio of skewness being not significantly different from 1 is not a relevant inference here.

Step 3: Conclusion.
The correct answers are (A) and (C), as they align with the interpretation of the \(F\)-distribution and the observed \(F\)-statistic.