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

$\mathrm{CCN}$ 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. At $a=.05$, what is your conclusion? d. Compute a $95 \%$ confidence interval for the population mean. Does it support your conclusion?

$\mathrm{CCN}$ 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. At $a=.05$, what is your conclusion?
d. Compute a $95 \%$ confidence interval for the population mean. Does it support your conclusion?

Key Concepts

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.

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.

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.

Transcript

00:01 So we're supposed to give what the hypotheses are for this question, and we would assume that the meantime of waiting in line is eight minutes.

00:09 And then, alternately, we want to know if it's different.

00:14 So we would just say not equal to eight.

00:17 And then we want to find what the p value is for this particular setting if we have the mean being 8.5 minutes from a sample size of 120 and we're to assume that the population standard deviation for wait time is 3.2 minutes and so we have to find that p value and let me draw a little picture right over here.

00:45 So again, we're going to be assuming that eight is right here.

00:50 That's what the mean is, and we get an 8.5.

00:55 Yeah, and because we're doing a not equal to a two tailed test, this means of 8.5 would be just as likely as a 7.5 so that shaded areas actually r p value.

01:07 So we're going to find what's the likelihood of getting an x bar that is greater than or equal to 8.5 or less than or equal to 7.5, which means we'll just double that.

01:20 And we need to convert that to a z value.

01:22 And it's going to be a z value because of our knowing the standard deviation of the population.

01:28 And so the test statistic will be 8.5 minus eight, divided by the standard deviation of the population over the square root of n, which is 120.

01:40 And let me get that in my calculator.

01:42 And let's see, we have 0.5 divided by left front to see 3.2 divided by the square root up 120 man, we get that test statistic to be 1.7116 and i'm going to use my normal cdf button to use.

02:10 To find that probability, you could as well use the table.

02:14 And i'm gonna use my lower bound as this number the upper bound as 1000, and i'm going to leave the mean and standard deviation at zero and one, and then i will find that value, and i'm going to have to then double it so times to to find the p value.

02:30 So i'm getting my p value 2.87 basically...

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