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?\no julians data shows that the recent seasons average jean price is not $75.\no julians data shows that he should order fewer jeans than before.\no julians data shows that the recent seasons average jean price is still $75.\no julians 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?\no julians data shows that the recent seasons average jean price is not $75.\no julians data shows that he should order fewer jeans than before.\no julians data shows that the recent seasons average jean price is still $75.\no julians 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 average price has changed). The sample mean $\bar{x}=68$, population standard - deviation $\sigma = 5$, and assume the sample size $n$ is large enough (since we have data over multiple seasons).

Step2: Calculate the z - score

The formula for the z - score in a one - sample z - test is $z=\frac{\bar{x}-\mu}{\frac{\sigma}{\sqrt{n}}}$. Although the sample size $n$ is not given, with a significant difference between $\bar{x}=68$ and $\mu = 75$ and a relatively small standard deviation $\sigma = 5$, the z - score will be a large value in magnitude. For a two - tailed test with a significance level of $\alpha=0.05$, the critical z - values are $z_{\alpha/2}=\pm 1.96$.

Step3: Make a decision

Since the sample mean of $68$ is far from the hypothesized mean of $75$ (a difference of $7$ with a standard deviation of $5$), we will reject the null hypothesis. 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 $75$.