julian is a manager at a clothing store for teens. he is analyzing the order for next season. data for the…

julian is a manager at a clothing store for teens. he is analyzing the order for next season. data for the previous 10 years suggests that teens are willing to spend an average of $75 for a pair of designer jeans with a standard deviation of $5. however, julian thinks the average may have changed due to a recession. he finds that the last three seasons of data show that teens spent an average of $68 on a pair of jeans. therefore, he performed a hypothesis test to see if the recent average is the same. julian used a significance level of 5% to perform the test. which of the following statements is valid based on the results?\n\njulians data shows that the recent seasons average jean price is not $75.\njulians data shows that he should order fewer jeans than before.\njulians data shows that the recent seasons average jean price is still $75.\njulians data is inconclusive, so he should order the same number of jeans.

julian is a manager at a clothing store for teens. he is analyzing the order for next season. data for the previous 10 years suggests that teens are willing to spend an average of $75 for a pair of designer jeans with a standard deviation of $5. however, julian thinks the average may have changed due to a recession. he finds that the last three seasons of data show that teens spent an average of $68 on a pair of jeans. therefore, he performed a hypothesis test to see if the recent average is the same. julian used a significance level of 5% to perform the test. which of the following statements is valid based on the results?\n\njulians data shows that the recent seasons average jean price is not $75.\njulians data shows that he should order fewer jeans than before.\njulians data shows that the recent seasons average jean price is still $75.\njulians data is inconclusive, so he should order the same number of jeans.

Answer

Explanation:

Step1: Identify the hypotheses

The null hypothesis $H_0:\mu = 75$ (the average price is still $75$) and the alternative hypothesis $H_1:\mu\neq75$. The sample mean $\bar{x}=68$, population mean $\mu_0 = 75$, population standard - deviation $\sigma = 5$.

Step2: Calculate the z - score

The formula for the z - score in a one - sample z - test is $z=\frac{\bar{x}-\mu_0}{\frac{\sigma}{\sqrt{n}}}$. Although the sample size $n$ is not given for the last three seasons data, assuming a reasonable sample size and using the fact that the sample mean of $68$ is quite far from the population mean of $75$ with a relatively small standard deviation of $5$. At a significance level of 5% (two - tailed test, critical values are approximately $z=\pm1.96$). A large difference between the sample mean and the population mean will likely lead to rejecting the null hypothesis.

Step3: Interpret the result

Since the sample mean of $68$ is much lower than the previous average of $75$, it is likely that the null hypothesis is rejected. This means that the recent seasons' average jean price is not $75$.

Answer:

Julian's data shows that the recent seasons' average jean price is not $75$.