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When statisticians talk about a point estimate, they are referring to a single value that serves as an estimate of a population parameter. In our exercise, the point estimate is the sample proportion of items returned at a store.

• To obtain the point estimate, you calculate the sample proportion, \( \hat{p} \).
• This is done by dividing the number of successes (returned items) by the total number of observations (total items sampled).

For the given problem, we've seen that the point estimate is calculated as \( \hat{p} = \frac{x}{n} \), where \( x = 12 \) (number of returned items) and \( n = 80 \) (total items sampled). Hence, the point estimate, or sample proportion, is \( \hat{p} = \frac{12}{80} = 0.15 \). This estimate tells us that around 15% of the sampled items were returned, which serves as a practical estimate of the overall return rate at the Houston store.

The sample proportion is a fundamental concept in statistical analysis, especially in scenarios like our exercise where you want to infer about a larger population based on a sample.

• It is represented by \( \hat{p} \) and calculated using the formula \( \hat{p} = \frac{x}{n} \).
• Here, \( x \) is the number of sampled observations with the desired attribute (e.g., returns), and \( n \) is the total sample size.

This value makes it easier to estimate other statistical measures, such as confidence intervals or conducting hypothesis tests. In our specific problem, the sample proportion is 0.15, informing us about the rate of returns in the sampled items. Since this is derived from the sample, it might vary from the actual population proportion due to sample variability. However, by understanding this number, decisions about inventory, customer service, and other store operations can be more informed.

Hypothesis testing is a vital statistical method used to determine if there's enough evidence to draw conclusions about a population parameter based on sample data. In our exercise, we're using hypothesis testing to compare the store's return rate against the national average.

• The null hypothesis \( H_0 \) suggests that the proportion of returns at the Houston store is the same as the national rate, i.e., 6% or 0.06.
• The alternative hypothesis \( H_a \) posits that the store's return rate differs from this national metric.

The hypothesis test uses a z-test for proportions to compute a test statistic. This involves calculating the standard error using the population proportion in the null hypothesis and comparing the computed z-value against the critical value from the z-distribution.In this exercise, with a calculated z-score of approximately 2.88 and a critical value of 1.96 for a 95% confidence level, we found that \(|2.88| > 1.96\). Thus, we can confidently reject the null hypothesis, concluding that the store's return rate is statistically different from the national average.