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A confidence interval is a range of values that estimates a population parameter based on sample data. It provides an upper and a lower limit within which we can say, with a certain level of confidence, that the true value of the parameter lies.
In hypothesis testing, especially when comparing proportions as in our exercise, constructing a confidence interval helps us understand the difference in proportions between two groups or samples.
For instance, when we say we have a 98% confidence interval for the difference in store return rates, it means we are 98% sure that the true difference between the two stores' return rates falls within this calculated range.
The confidence interval for the difference in proportions is calculated using the formula: \[(p1 - p2) \pm Z_{\alpha/2} \times SE\] Where:

• \(p1, p2\) are the sample proportions
• \(Z_{\alpha/2}\) is the z-value corresponding to the desired confidence level
• \(SE\) is the standard error calculated for these proportions

This ensures we incorporate both sampling variation and desired confidence level when estimating the parameter range.

Proportions represent the fraction of the total sample that holds a particular characteristic. In the context of our exercise, we are looking at the proportion of sales with returned items at each department store.
Calculating these proportions involves dividing the number of favorable outcomes by the total number of events. So for Store A, the proportion is: - \(\text{Proportion A} = \frac{280}{800} = 0.35\)
Similarly, for Store B, the calculation is: - \(\text{Proportion B} = \frac{279}{900} = 0.31\)
Using proportions provides a more standardized way to compare different groups since it accounts for varied sample sizes. Whether using smaller or larger samples, proportions allow us to see the ratio of occurrences in a consistent manner.

The standard error (SE) is a measure of the variability or dispersion of sample statistics from the true population parameter. In simpler terms, it shows how much the sample proportion could vary from the actual population proportion due to sampling errors.
For proportions, the formula for standard error is: \[SE = \sqrt{\frac{p1(1-p1)}{n1} + \frac{p2(1-p2)}{n2}}\]Where:

• \(p1, p2\) are the sample proportions of Store A and Store B
• \(n1, n2\) are the respective sample sizes of Store A and Store B

The calculated SE helps us understand the precision of our estimate when comparing two proportions.
In our solution, the SE was found to be approximately 0.03075, indicating the likely fluctuation in the sample difference of return rates due to sampling randomness.

The significance level, often denoted as \(\alpha\), represents the probability of rejecting the null hypothesis when it is actually true. It is the benchmark for deciding whether observed outcomes are statistically significant or just due to chance.
In our problem, a 1% significance level was used, meaning there is a 1% risk of concluding that the proportion of returned sales in Store A is higher than Store B when it is not.
This critical value derived from the significance level is crucial in hypothesis testing as it dictates the threshold for our test statistic. If the test statistic exceeds the critical value, we reject the null hypothesis.
However, in this exercise, the test statistic was 1.30, which is less than the critical value of 2.33 at the 1% significance level, leading us to fail to reject the null hypothesis.