CCN and ActMedia provided a television channel targeted to...

CCN and ActMedia provided a television channel targeted to individuals waiting in supermarket checkout lines. The channel showed news, short features, and advertisements. The length of the program was based on the assumption that the population mean time a shopper stands in a supermarket checkout line is 8 minutes. A sample of actual waiting times will be used to test this assumption and determine whether actual mean waiting time differs from this standard. a. Formulate the hypotheses for this application. b. A sample of 120 shoppers showed a sample mean waiting time of 8.5 minutes. Assume a population standard deviation of $\sigma=3.2$ minutes. What is the $p$ -value? c. $\quad$ At $a=.05,$ what is your conclusion? d. Compute a $95 \%$ confidence interval for the population mean. Does it support your conclusion?

CCN and ActMedia provided a television channel targeted to individuals waiting in supermarket checkout lines. The channel showed news, short features, and advertisements. The length of the program was based on the assumption that the population mean time a shopper stands in a supermarket checkout line is 8 minutes. A sample of actual waiting times will be used to test this assumption and determine whether actual mean waiting time differs from this standard.
a. Formulate the hypotheses for this application.
b. A sample of 120 shoppers showed a sample mean waiting time of 8.5 minutes. Assume a population standard deviation of $\sigma=3.2$ minutes. What is the $p$ -value?
c. $\quad$ At $a=.05,$ what is your conclusion?
d. Compute a $95 \%$ confidence interval for the population mean. Does it support your conclusion?

Key Concepts

Confidence Interval

A confidence interval provides a range of plausible values for the population parameter, calculated from the sample data. A 95% confidence interval means that if the sampling process were repeated many times, approximately 95% of the computed intervals would contain the true population mean. Confidence intervals help to assess the precision and uncertainty of the estimated parameter, and they can also be compared with the null hypothesis value to support the decision from hypothesis testing.

Hypothesis Testing

This concept involves formulating a null hypothesis, which represents the status quo or the assumption of no effect, and an alternative hypothesis, which represents the claim being tested for in the sample data. In the context of comparing a sample mean to a specific value, the null hypothesis typically states that the population mean is equal to the specified value, while the alternative hypothesis states that it is different.

Z-Test for Means

When the population standard deviation is known, the Z-test is used to evaluate the difference between the sample mean and the hypothesized population mean. The test statistic is calculated by standardizing the difference using the known standard deviation, and it follows a standard normal distribution under the null hypothesis.

p-value

The p-value is the probability, computed under the null hypothesis, of obtaining a test statistic as extreme or more extreme than the one observed. It serves as a measure of the evidence against the null hypothesis; a smaller p-value indicates stronger evidence to reject the null hypothesis.

Significance Level

The significance level, commonly denoted as alpha (?), is the threshold used to decide whether to reject the null hypothesis. If the p-value is lower than the significance level, the result is considered statistically significant, leading to rejection of the null hypothesis. Otherwise, the null hypothesis is not rejected.

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