Question

F Test Statistic

a. If \(s_1^2\)represents the larger of two sample variances, can the F test statistic ever be less than 1?

b. Can the F test statistic ever be a negative number?

c. If testing the claim that \(\sigma _1^2 \ne \sigma _2^2\), what do we know about the two samples if the test statistic F is very close to 1?

d. Is the F distribution symmetric, skewed left, or skewed right?

Short answer

a.No. If \(s_1^2\)is greater than \(s_2^2\), then the F statistic cannot be less than 1.

b.No. Fstatistic cannot be a negative number.

c.When the value of the F statistic is very close to 1, it means that the two samples have equal variances.

d.The F-distribution is skewed to the right or positively skewed.

Step-by-step solution

Given Information

Let,

\(s_1^2\)denote thelarger of the two sample variances.

\(s_2^2\)denote theSmaller of the two sample variances

\(\sigma _1^2\)denote theLarger variance of the population.

\(\sigma _2^2\)denote the population variance of the other population.

Value of the F statistic

a.

The expression of the F statistic is written as follows:

\(F = \frac{{s_1^1}}{{s_2^2}}\)

If\(s_1^2\)is greater than\(s_2^2\), the value of the F statistic is expressed as follows:

\(\begin{array}{c}F = \frac{{s_1^1}}{{s_2^2}}\\ > 1\end{array}\)

Hence, the F test statistic is greater than one when \(s_1^2\)has a larger variance.

No, the F stataistic cannot be less than 1.

Range of F - distribution

b.

The F-distribution is the ratio of the squares of the standard deviations of the two samples.

It is known that the square of a number cannot be less than 0.

Hence, the ratio of two positive numbers will also be a positive number.

Therefore, the value of the F statistic can never be negative.

Interpreting the value of the F statistic

c.

The following null hypothesis is considered:

Null Hypothesis: The population variances are not equal.

Symbolically,

\({H_0}:\sigma _1^2 \ne \sigma _2^2\)

If the F statistic is close to 1, it implies that the values of the sample variances are approximately equal because the F statistic is the ratio of the two sample variances.

Thus, if the F-statistic is close to 1, the sample variances are approximately equal.

Symmetricity of F-distribution

d.

The F-distribution is positively skewed or skewed to the right and not symmetric.

A distribution that is skewed to the right has a tail towards the right side.

Also, the F-distribution has a mean value that is greater than the medina value and hence, the distribution is skewed to the right.