a real estate agent is working for a developer who claims that the average commute time to downtown is 20…
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?\nthe z - statistic is 0.22, so the null hypothesis should not be rejected.\nthe z - statistic is 1.69, so the null hypothesis should not be rejected.\nthe z - statistic is 3.32, so the null hypothesis should be rejected.\nthe z - statistic is 3.67, so the null hypothesis should be rejected.
Answer
Explanation:
Step1: Identify the formula for z - statistic
The formula for the z - statistic in a one - sample z - test for the mean 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.
Step2: Substitute the given values
We are given that $\mu = 20$, $\sigma=7$, $n = 15$, and $\bar{x}=26$. $z=\frac{26 - 20}{\frac{7}{\sqrt{15}}}$ First, calculate the denominator $\frac{7}{\sqrt{15}}\approx\frac{7}{3.873}\approx1.807$. Then, calculate the numerator $26 - 20=6$. So, $z=\frac{6}{1.807}\approx3.32$.
Step3: Determine the decision rule
For a significance level of 5% in a two - tailed test, the critical z - values are approximately $\pm1.96$. If $|z|>1.96$, we reject the null hypothesis. Since $z = 3.32>1.96$, we reject the null hypothesis.
Answer:
The z - statistic is 3.32, so the null hypothesis should be rejected.