91Ó°ÊÓ

The sample proportion is a key concept when analyzing data collected from a survey or experiment. In this context, it represents the fraction of individuals within a sample that exhibit a particular trait or behavior, such as not sending in a rebate claim form. To calculate the sample proportion, divide the number of individuals with the trait by the total number of individuals in the sample.

For example, if 250 out of 1000 consumers did not return their rebate form, the sample proportion is calculated as \( \hat{p} = \frac{250}{1000} = 0.25 \).

Understanding the sample proportion enables researchers to make predictions and draw conclusions about a larger population. It serves as a foundation for further statistical calculations such as confidence intervals or hypothesis tests.

The standard error (SE) measures the potential variability of a sample statistic from the true population parameter due to random sampling. It quantifies how much a sample proportion, like our \(\hat{p} = 0.25\), could differ from the actual population proportion due to chance.

In this exercise, the formula for the standard error of the sample proportion \( SE \) is given by: \[ SE = \sqrt{\frac{\hat{p}(1-\hat{p})}{n}} \] where \( \hat{p} \) is the sample proportion and \( n \) is the sample size.

For the consumer rebate scenario, with \( \hat{p} = 0.25 \) and \( n = 1000 \), the standard error is calculated as: \[ SE = \sqrt{\frac{0.25 \times 0.75}{1000}} \approx 0.0137 \]

A smaller standard error suggests a more accurate sample estimate of the population parameter, while a larger standard error suggests less precision.

The Z-score represents how many standard deviations a data point is from the mean of a distribution. In hypothesis testing, Z-scores are used to determine the likelihood of observing a result given a standard normal distribution.

For confidence intervals, the Z-score represents the critical value for a given confidence level. For example, a common confidence level is 95%, and the Z-score corresponding to 95% confidence is approximately 1.645.

This Z-score is used in conjunction with the standard error to compute confidence bounds around a sample estimate. The Z-score allows us to understand the probability of finding the observed sample proportion (or more extreme values) assuming the null hypothesis is true.

Hypothesis testing is a method to make decisions or inferences about population parameters based on sample data. It involves setting up two opposing hypotheses: the null hypothesis \(H_0\) and the alternative hypothesis \(H_a\).

In this exercise, you are assessing whether the true proportion of consumers who skip sending in rebate forms is less than \(\frac{1}{3}\). Your null hypothesis might state that the true proportion is \(\geq \frac{1}{3}\), while your alternative hypothesis suggests it's \(< \frac{1}{3}\).

• Null hypothesis: \( H_0: p \geq \frac{1}{3} \)
• Alternative hypothesis: \( H_a: p < \frac{1}{3} \)

By calculating the upper confidence bound of the sample proportion and comparing it to \(\frac{1}{3}\), you can determine if there is compelling evidence to reject the null hypothesis in favor of the alternative. If the upper bound is less than \(\frac{1}{3}\), as in this case, you have sufficient evidence to support the claim that fewer consumers send in rebate forms than \(\frac{1}{3}\).