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A company that has many department stores in the southern states wanted to find at two such stores the percentage of sales for which at least one of the items was returned. A sample of 800 sales randomly selected from Store A showed that for 280 of them at least one item was returned. Another sample of 900 sales randomly selected from Store B showed that for 279 of them at least one item was returned. a. Construct a 98\% confidence interval for the difference between the proportions of all sales at the two stores for which at least one item is returned. b. Using a \(1 \%\) significance level, can you conclude that the proportions of all sales for which at least one item is returned is higher for Store A than for Store \(\mathrm{B}\) ?

Step by step solution

01

Calculate Sample Proportions

For store A, the sample proportion (p_A) of sales with at least one returned item is 280/800 = 0.35. For store B, the sample proportion (p_B) of sales with at least one returned item is 279/900 = 0.31.

02

Construct the Confidence Interval

To construct a 98\% confidence interval for the difference between the proportions (p_A - p_B), we first find the standard error (SE) of the difference using the formula: SE = sqrt[(p_A*(1-p_A)/n_A) + (p_B*(1-p_B)/n_B)]. Here, n_A=800 and n_B=900. The critical value (z) for a 98\% confidence interval can be found from the standard normal table, which is approximately 2.33. The confidence interval is thus (p_A - p_B) ± z*SE.

03

Perform the Hypothesis Test

The null hypothesis is 'There is no difference between the proportions of all sales where at least one item is returned between store A and store B'. The alternative hypothesis is 'The proportion of all sales where at least one item is returned is higher for store A than for store B'. We find the test statistic using the formula: z = (p_A - p_B - 0)/SE. The p-value is then the area under the normal curve to the right of this z-score. If p-value < 0.01 (significance level), we reject the null hypothesis.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Sample Proportions

Sample proportions give us a way to understand a part of a whole by looking at just a sample of it. In this case, for Store A, 280 out of 800 sales had items returned, giving us a sample proportion for Store A as \( p_A = \frac{280}{800} = 0.35 \). Similarly, for Store B, 279 out of 900 sales had items returned, providing Store B's sample proportion as \( p_B = \frac{279}{900} = 0.31 \). This helps us summarize significant data about the two stores' sales.

• Sample proportions are often expressed in decimal form.
• They provide insight into the behavior of larger populations.

Understanding sample proportions allows us to make statistical inferences about the whole group from which the sample was drawn.

Hypothesis Testing

Hypothesis testing is a method to determine if there's enough evidence to support a particular belief about a parameter. In our scenario, we want to test if Store A has a higher proportion of returns than Store B. The null hypothesis \( H_0 \) states that there is no difference in return proportions between the two stores. In contrast, the alternative hypothesis \( H_a \) suggests that the proportion is higher for Store A. The test involves computing a test statistic, which helps in making a decision.

• If the test statistic is significant, we might reject the null hypothesis.
• Decisions are made based on a comparison of the calculated p-value with a pre-determined significance level.

Hypothesis testing helps in making data-driven decisions with a defined level of confidence.

Significance Level

The significance level, often denoted by \( \alpha \), is a threshold used to determine whether a hypothesis test result is statistically significant. In our case, a 1% significance level \( (\alpha = 0.01) \) is used.Using a lower significance level means we're looking for stronger evidence to reject the null hypothesis. For instance, with \( \alpha = 0.01 \), we only reject the null hypothesis if there's less than a 1% chance that the observed data could occur if the null hypothesis were true.

• Common significance levels are 0.05 (5%) and 0.01 (1%).
• A lower significance level reduces the risk of a Type I error (incorrectly rejecting a true null hypothesis).

The significance level helps control the confidence with which we make statistical inferences.

Standard Error

Standard error provides a measure of variability or deviation from the mean in a sample distribution. For proportions, it helps in constructing confidence intervals and hypothesis tests. In our example, the standard error for the difference in proportions is calculated as:\[SE = \sqrt{\left( \frac{p_A(1-p_A)}{n_A} \right) + \left( \frac{p_B(1-p_B)}{n_B} \right)}\]Here, \( n_A = 800 \) and \( n_B = 900 \) are the sample sizes for stores A and B respectively.

• Standard error decreases with larger sample sizes.
• It helps determine how much the sample mean is expected to vary from the actual population mean.

Understanding standard error allows us to estimate the precision of the sample proportions.