1. Returns for the S&P 500 average 6% annually historically. Morning Star wants to perform a hypothesis test to determine if a particular mutual fund has outperformed the S&P 500. Use the sheet “Return” to perform a test to determine if the mutual fund average return is greater than
## 
##  One Sample t-test
## 
## data:  Return
## t = 5.1, df = 14, p-value = 0.00008
## alternative hypothesis: true mean is greater than 6
## 95 percent confidence interval:
##  8.793   Inf
## sample estimates:
## mean of x 
##     10.27
  1. Describe the Type I error associated with this problem.

The Type I error for this problem would be suggesting the mutual fund performed better than the S&P 500 when it really doesn’t.

b.Specify the null and alternative hypothesis for this test.

\[ \begin{align*} H_{0}: \mu = 6\% \\ H_{a}: \mu > 6\% \end{align*} \]

  1. What is the result of the test. Report the test statistic and the p-value. The test statistic is 5.1 with a p-value of approximately equal to 0. Reject the null in favor of the alternative. This suggests that the mutual fund has outperformed the S&P 500.
  1. An email promotion is considered to be successful if 25% of the recipients make a purchase. A sample of 50 customers resulted in a sample proportion of 28%. Perform a hypothesis test to determine if the population proportion is greater than 25%. Report the p-value and the test statistic related to the test. Clearly interpret the result of the test.

\[\begin{align*} H_{0}: p = 25\% \\ H_{a}: p > 25\% \end{align*} \]

The normality assumption holds since \(n*p > 10\).

The results would be:

Test Stat = 0.49

p-value = 0.312

Assuming a 0.05 level of alpha, do not reject the null in favor of the alternative. There is not enough information to suggest that the proportion of customers who made a purchase is greater than 25%. It seems as if the promotion was not a success.

  1. A store wants to determine if sales between two grocery stores are different to characterize the different markets. Use the worksheet grocery to answer the following questions. Sales are measured in 1,000’s of dollars.
## 
##  Welch Two Sample t-test
## 
## data:  grocery_1 and grocery_2
## t = -1.9, df = 38, p-value = 0.07
## alternative hypothesis: true difference in means is not equal to 0
## 95 percent confidence interval:
##  -11.2318   0.3652
## sample estimates:
## mean of x mean of y 
##     91.53     96.97

a.Specify the null and alternative hypothesis for this test.

\[\begin{align*} H_{0}: \mu_{1} = \mu_{2} \\ H_{a}: \mu_{1} \neq \mu_{2} \end{align*} \]

  1. What is the result of the test. Report the test statistic and the p-value.

The test-stat is -1.90 with a p-value of 0.065. Using an alpha of 0.05, do not reject the null in favor of the alternative.