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

Answer:

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:

$\chi^2 =\frac{n-1}{\sigma^2_0} s^2$

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

$\chi^2 =\frac{136-1}{17500^2} 18500^2 =150.869$

Step-by-step 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:

$\chi^2 =\frac{n-1}{\sigma^2_0} s^2$

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

$\chi^2 =\frac{136-1}{17500^2} 18500^2 =150.869$