Question

A statement from an official report said that Realtors make $85,000 a year (national mean). Researcher thinks that Realtors in the Des Moines area have a mean annual salary of more than 85,000 a year. At alpha = 0.05, sigma = $12,549, n = 36 and sample mean = $88,595, can we conclude that realtors in the Des Moines area make more than $85,000? Sample Salaries $98,356.00 $120,492.00 $119,398.00 $92,682.00 $113,284.00 $79,398.00 $82,503.00 $81,485.00 $84,083.00 $76,384.00 $86,539.00 $69,894.00 $67,810.00 $84,219.00 $90,242.00 $104,294.00 $86,642.00 $85,646.00 $73,286.00 $65,074.00 $77,829.00 $106,881.00 $83,221.00 $88,483.00 $80,998.00 $84,734.00 $126,949.00 $64,085.00 $81,469.00 $85,769.00 $98,864.00 $92,132.00 $65,025.00 $101,594.00 $91,250.00 $98,426.00

Answer 1

Answer:

Here, the sample size is 36 (greater than 30) and the population standard deviation () is known, so z test can be used for testing the hypothesis. The following steps of hypothesis testing can be performed as below:

Step 1: Set the null and alternative hypotheses

In this example, the researcher is trying to verify whether there is evidence to support the claim that Realtors in the Des Moines area have a mean annual salary of more than $85,000 a year at a 0.05 significance level.

Null Hypothesis:

Alternative Hypothesis:

Step 2: Appropriate test statistic

z formula for single population mean and for known population standard deviation is given as

Step 3: Set the decision rule

Here, the level of significance = 0.05. The critical z value using standard table is z_0.05 = 1.96. If the computed test statistic is less than 1.96, then the decision is to reject the null hypothesis and accept the alternative hypothesis.

Step 4: Calculation for test statistic

Here, n = 36, , , and the hypothesized mean

By substituting all the values in the above given formula, we get:

Step 5: Statistical Conclusion

The calculated z value s 1.7189, which is less than 1.96. Hence, the statistical conclusion is to reject the null hypothesis and accept the alternative hypothesis at 0.05 level of significance. On the basis of hypothesis testing, it can be concluded that the Realtors in the Des Moines area have a mean annual salary of more than $85,000 a year at a 0.05 significance level.