Question

​Claim: High school teachers have incomes with a standard deviation that is more than ​$17, 500. A recent study of 136 high school teacher incomes showed a standard deviation of ​$18, 500.

A. Express the original claim in symbolic form.
B. Identify the null and the alternative hypotheses that should be used to arrive at the conclusion that supports the claim.

Answer 1

Part a

For this case the claim is:

`\sigma >17500`

And that represent the alternative hypothesis.

Part b: Null and alternative hypothesis

On this case we want to check if the population deviation is higher than 17500, so the system of hypothesis would be:

Null Hypothesis:
`\sigma \leq 17500`

Alternative hypothesis:
`\sigma >17500`

Calculate the statistic

For this test we can use the following statistic:

`\chi^2 =(n-1)/(\sigma^2_0) s^2`

And this statistic is distributed chi square with n-1 degrees of freedom. We have eveything to replace.

`\chi^2 =(136-1)/(17500^2) 18500^2 =150.869`

Explanation:

Notation and previous concepts

A chi-square test is "used to test if the variance of a population is equal to a specified value. This test can be either a two-sided test or a one-sided test. The two-sided version tests against the alternative that the true variance is either less than or greater than the specified value"

`n=136` represent the sample size

`\alpha` represent the confidence level

`s^2 =18500^2` represent the sample variance obtained

`\sigma^2_0 =17500^2` represent the value that we want to test

Part a

For this case the claim is:

`\sigma >17500`

And that represent the alternative hypothesis.

Part b: Null and alternative hypothesis

On this case we want to check if the population deviation is higher than 17500, so the system of hypothesis would be:

Null Hypothesis:
`\sigma \leq 17500`

Alternative hypothesis:
`\sigma >17500`

Calculate the statistic

For this test we can use the following statistic:

`\chi^2 =(n-1)/(\sigma^2_0) s^2`

And this statistic is distributed chi square with n-1 degrees of freedom. We have eveything to replace.

`\chi^2 =(136-1)/(17500^2) 18500^2 =150.869`