Questions: The marketing manager of a firm that produces laundry products decides to test market a new laundry product in each of the firm's two sales regions. He wants to determine whether there will be a difference in mean sales per market per month between the two regions. A random sample of 12 supermarkets from Region 1 had mean sales of 78.7 with a standard deviation of 6.7. A random sample of 16 supermarkets from Region 2 had a mean sales of 86.3 with a standard deviation of 7. Does the test marketing reveal a difference in potential mean sales per market in Region 2? Let μ1 be the mean sales per market in Region 1 and μ2 be the mean sales per market in Region 2. Use a significance level of α=0.1 for the test. Assume that the population variances are not equal and that the two populations are normally distributed. Step 1 of 4: State the null and alternative hypotheses for the test. H0: μ1 = μ2 Ha: μ1 ≠ μ2

The marketing manager of a firm that produces laundry products decides to test market a new laundry product in each of the firm's two sales regions. He wants to determine whether there will be a difference in mean sales per market per month between the two regions. A random sample of 12 supermarkets from Region 1 had mean sales of 78.7 with a standard deviation of 6.7. A random sample of 16 supermarkets from Region 2 had a mean sales of 86.3 with a standard deviation of 7. Does the test marketing reveal a difference in potential mean sales per market in Region 2? Let μ1 be the mean sales per market in Region 1 and μ2 be the mean sales per market in Region 2. Use a significance level of α=0.1 for the test. Assume that the population variances are not equal and that the two populations are normally distributed. Step 1 of 4: State the null and alternative hypotheses for the test. H0: μ1 = μ2 Ha: μ1 ≠ μ2
Transcript text: The marketing manager of a firm that produces laundry products decides to test market a new laundry product in each of the firm's two sales regions. He wants to determine whether there will be a difference in mean sales per market per month between the two regions. A random sample of 12 supermarkets from Region 1 had mean sales of 78.7 with a standard deviation of 6.7 . A random sample of 16 supermarkets from Region 2 had a mean sales of 86.3 with a standard deviation of 7. Does the test marketing reveal a difference in potential mean sales per market in Region 2? Let $\mu_{1}$ be the mean sales per market in Region 1 and $\mu_{2}$ be the mean sales per market in Region 2. Use a significance level of $\alpha=0.1$ for the test. Assume that the population variances are not equal and that the two populations are normally distributed. Step 1 of 4: State the null and alternative hypotheses for the test. Answer 2 Points Tables Keypad Keyboard Shortcuts $H_{0}: \mu_{1}$ $\square$ $\mu_{2}$ $H_{a}: \mu_{1}$ $\square$ $\mu_{2}$
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Solution

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Solution Steps

To determine whether there is a difference in mean sales per market per month between the two regions, we need to perform a hypothesis test. The null hypothesis ($H_0$) will state that there is no difference in mean sales between the two regions, while the alternative hypothesis ($H_a$) will state that there is a difference.

Solution Approach
  1. Define the null hypothesis ($H_0$) and the alternative hypothesis ($H_a$).
  2. Use a two-sample t-test for means with unequal variances to test the hypotheses.
  3. Calculate the test statistic and the p-value.
  4. Compare the p-value with the significance level $\alpha=0.1$ to make a decision.
Step 1: State the Hypotheses

We need to determine whether there is a difference in mean sales per market per month between the two regions. We define the null and alternative hypotheses as follows:

  • Null hypothesis (\(H_0\)): \(\mu_1 = \mu_2\)
  • Alternative hypothesis (\(H_a\)): \(\mu_1 \neq \mu_2\)
Step 2: Calculate the Test Statistic and p-value

Given the sample data:

  • Region 1: \(\bar{x}_1 = 78.7\), \(s_1 = 6.7\), \(n_1 = 12\)
  • Region 2: \(\bar{x}_2 = 86.3\), \(s_2 = 7.0\), \(n_2 = 16\)

We perform a two-sample t-test for means with unequal variances. The test statistic and p-value are calculated as follows:

  • Test statistic (\(t\)): \(-2.9138\)
  • p-value: \(0.0075\)
Step 3: Compare p-value with Significance Level

The significance level is \(\alpha = 0.1\). We compare the p-value to \(\alpha\):

  • Since \(0.0075 < 0.1\), we reject the null hypothesis.

Final Answer

Based on the test statistic and p-value, we reject the null hypothesis. This indicates that there is a significant difference in mean sales between the two regions.

\[ \boxed{\text{Reject the null hypothesis: There is a significant difference in mean sales between the two regions.}} \]

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