Question

A researcher claims that the variation in the salaries of elementary school teachers is greater than the variation in the salaries of secondary school teachers (the claim). A random sample of 30 elementary school teachers has a variance of $ 8324, and a random sample of 30 secondary school teachers has a variance of $2862. At α= 0.05, can the researcher conclude that the variation in the elementary school teachers' Salaries is greater than the variation in the secondary teachers' salaries? Use the P-value method.

Assume that all variables are normally distributed.

a. State the hypotheses and identify the claim.
b. Find the critical value.
c. Compute the test value.
d. Make the decision.
e. Summarize the results.

Answer 1

a

The null hypothesis is
`H_o : \sigma^2_1 = \sigma^2 _2`

The alternative hypothesis is
`H_a : \sigma_1 ^2 > \sigma^2_2`

b

`F_(critical) = 1.8608`

c

`F = 2.9085`

d

The decision rule is

Reject the null hypothesis

e

There is sufficient evidence to support the researchers claim

Explanation:

From the question we are told that

The first sample size is
`n_1 = 30`

The sample variance for elementary school is
`s^2_1 = 8324`

The second sample size is
`n_2 = 30`

The sample variance for the secondary school is
`s^2_2 = 2862`

The significance level is
`\alpha = 0.05`

The null hypothesis is
`H_o : \sigma^2_1 = \sigma^2 _2`

The alternative hypothesis is
`H_a : \sigma_1 ^2 > \sigma^2_2`

Generally from the F statistics table the critical value of
`\alpha = 0.05` at first and second degree of freedom
`df_1 = n_1 - 1 = 30 - 1 = 29` and
`df_2 = n_2 - 1 = 30 - 1 = 29` is

`F_(critical) = 1.8608`

Generally the test statistics is mathematically represented as

`F = (s_1^2 )/(s_2^2)`

=>
`F = (8324 )/(2862)`

=>
`F = 2.9085`

Generally from the value obtained we see that
`F > F_(critical )` Hence

The decision rule is

Reject the null hypothesis

The conclusion is

There is sufficient evidence to support the researchers claim