To analyze this hypothesis test, we will proceed step-by-step.

Given Data

The differences between Store 1 and Store 2's sales ($d = \text{Store 1} - \text{Store 2}$) for the 12 days are:

$d = [-44, -161, -87, -52, -196, -52, -149, -244, -156, -170, 100, 11]$

The hypothesis test involves the population mean difference ($\mu_d$).

(a) Hypotheses

We are testing whether the mean daily sales of Store 2 exceeds that of Store 1. Since the differences are calculated as $\text{Store 1} - \text{Store 2}$:

• Null Hypothesis ($H_0$): $\mu_d = 0$ (no difference in the mean sales).
• Alternative Hypothesis ($H_1$): $\mu_d < 0$ (Store 2 has higher mean sales).

(b) Type of Test Statistic

Since we are comparing means and the population of differences is assumed to be normally distributed, we use a t-test for the paired differences.

(c) Compute the Test Statistic

1. Calculate the mean difference ($\bar{d}$): $\bar{d} = \frac{\sum d}{n}$ $\bar{d} = \frac{-44 - 161 - 87 - 52 - 196 - 52 - 149 - 244 - 156 - 170 + 100 + 11}{12}$ $\bar{d} = \frac{-1200}{12} = -100$

2. Calculate the standard deviation of the differences ($s_d$): $s_d = \sqrt{\frac{\sum (d_i - \bar{d})^2}{n - 1}}$ Calculate $(d_i - \bar{d})^2$ for each difference, sum them, and divide by $n-1$:

   • $s_d \approx 95.394$

3. Calculate the test statistic: $t = \frac{\bar{d} - \mu_0}{s_d / \sqrt{n}}$ $t = \frac{-100 - 0}{95.394 / \sqrt{12}}$ $t \approx \frac{-100}{27.533} \approx -3.631$

(d) Critical Value

For a one-tailed test at $\alpha = 0.10$, with $n-1 = 11$ degrees of freedom, the critical value of $t$ is approximately: $t_{critical} = -1.363$

(e) Conclusion

Since $t = -3.631$ is less than $t_{critical} = -1.363$, we reject the null hypothesis ($H_0$).

Final Answer

At the 0.10 level of significance, there is enough evidence to conclude that the mean daily sales of Store 2 exceed that of Store 1.

Questions for Further Exploration

1. How does changing the significance level affect the conclusion?
2. What assumptions are necessary for the validity of this test?
3. What if the differences were not normally distributed—what test would be appropriate?
4. How can the owner use this data to estimate the impact of the improvements on sales?
5. What other variables could impact sales performance?

Tip:

Always verify the assumptions of the test, such as normality, to ensure valid results.