Question

a real estate agent is working for a developer who claims that the average commute time to downtown is 20 minutes with a standard deviation of 7 minutes. stephon is an independent real estate agent and wants to check the times for his client. he took a random sample of 15 commute times and found an average of 26 minutes. he did hypothesis testing using a significance level of 5%. which conclusion could he make?
the z - statistic is 0.22, so the null hypothesis should not be rejected.
the z - statistic is 1.69, so the null hypothesis should not be rejected.
the z - statistic is 3.32, so the null hypothesis should be rejected.
the z - statistic is 3.67, so the null hypothesis should be rejected.

Explanation:

Step1: Recall z - statistic formula for hypothesis testing

The formula for the z - statistic in a one - sample z - test (when the population standard deviation \(\sigma\) is known) is \(z=\frac{\bar{x}-\mu}{\frac{\sigma}{\sqrt{n}}}\), where \(\bar{x}\) is the sample mean, \(\mu\) is the population mean, \(\sigma\) is the population standard deviation, and \(n\) is the sample size.

Given: \(\mu = 20\) minutes, \(\bar{x}=26\) minutes, \(\sigma = 7\) minutes, \(n = 15\)

Step2: Calculate the z - statistic

First, calculate the standard error \(SE=\frac{\sigma}{\sqrt{n}}=\frac{7}{\sqrt{15}}\approx\frac{7}{3.87298}\approx1.807\)

Then, calculate the z - statistic: \(z=\frac{\bar{x}-\mu}{SE}=\frac{26 - 20}{1.807}=\frac{6}{1.807}\approx3.32\)

Step3: Determine the critical value and make a decision

For a significance level of \(\alpha = 0.05\) (two - tailed test, but if it is a one - tailed test, the critical value for \(\alpha=0.05\) is \(z_{\alpha}=1.645\); for two - tailed, it is \(z_{\alpha/2}=1.96\)). Our calculated z - statistic is \(z = 3.32\). Since \(3.32>1.96\) (for two - tailed) or \(3.32 > 1.645\) (for one - tailed), we reject the null hypothesis.

Now let's check the options:

• Option 1: z - statistic is 0.22 (incorrect calculation), so this is wrong.
• Option 2: z - statistic is 1.69 (incorrect calculation), so this is wrong.
• Option 3: z - statistic is 3.32, and since \(3.32\) is greater than the critical value (1.645 for one - tailed or 1.96 for two - tailed at \(\alpha = 0.05\)), we reject the null hypothesis. So we reject the null hypothesis. This option is correct.
• Option 4: z - statistic is 3.67 (incorrect calculation), so this is wrong.

Answer:

The z - statistic is 3.32, so the null hypothesis should be rejected. (Option: The z - statistic is 3.32, so the null hypothesis should be rejected.)