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 8.4 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 8.4 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 is a range of values calculated from the sample data that is likely to contain the true population parameter with a certain level of confidence (commonly 95%). The interval provides a measure of the precision of the sample estimate and is used to assess the plausibility of different values of the parameter, thereby supporting conclusions drawn from hypothesis testing.

Significance Level

The significance level, often denoted by alpha, is a threshold used in hypothesis testing to determine whether the p-value is small enough to reject the null hypothesis. Common levels include 0.05 and 0.01. If the p-value falls below the significance level, the observed data are considered statistically significant, implying that the evidence is strong enough to conclude that the effect or difference is not due to random chance.

Hypothesis Testing

Hypothesis testing involves stating a null hypothesis, which represents a baseline or assumed value, and an alternative hypothesis, which represents a claim of interest. In testing a population mean, the null hypothesis typically posits that the mean is equal to a specific value, while the alternative hypothesizes that the mean is different. This framework helps in making decisions about whether observed sample statistics are consistent with the assumed population parameter.

One-Sample Z-Test

A one-sample Z-test is used when testing a hypothesis about the population mean with a known population standard deviation. It involves computing a test statistic that compares the difference between the sample mean and the hypothesized mean relative to the variability expected in the sample. This test statistic is then referenced against the standard normal distribution to determine the likelihood of observing such data if the null hypothesis were true.

p-Value

The p-value is the probability of obtaining a sample statistic as extreme as, or more extreme than, the observed value under the assumption that the null hypothesis is true. It quantifies the evidence against the null hypothesis: a smaller p-value suggests stronger evidence, potentially leading to its rejection. In hypothesis testing, this value is compared to a pre-established significance level to decide whether the results are statistically significant.

Transcript

00:01 So we're looking at the mean wait time, and we are going to be assuming that the mean wait time is eight minutes, and alternately that it's different.

00:15 So that's part a.

00:18 Now, they take a sample, and we know that the sample is 120 people, and we know that the x bar they get for that is 8 .4 minutes.

00:29 We are to assume that the population standard deviation is 3 .2 minutes.

00:37 Now, regardless of the shape of the distribution, and let me just do this, regardless of the shape of the distribution, the sampling distribution is going to be approximately normal because of the central limit theorem.

00:50 And the central limit theorem, if we take, we're going to assume that this is the mean is 8, and we're going to have that the standard deviation of the x bars all coming from a sample sizes of 120 is equivalent to 3 .2 divided by the square root of 120.

01:07 So we want to find what's the likelihood if the mean is actually 8.

01:14 Excuse me, 8, that we would get an 8 .4 or higher, or this is 0 .4, 7 .6 or lower.

01:24 And these two together will be our p value.

01:27 So let's find that probability.

01:28 So what's the probability of getting an x bar that's greater than or equal to 8 .4 in our picture? and then we'll double it since we want both ends for our p value.

01:39 And we'll convert this to a z because we know our population standard deviation.

01:45 And so we have 8 .4 minus 8 divided by that 3 .2 over the square root of 120.

01:52 And be careful about the way you put that in your calculator.

01:55 I know my numerator is 0 .4 and then divided by left -hand parentheses, 3 .2 divided by the square root of 120.

02:06 And let me get that answer.

02:09 And i get that that z value comes out to be 1 .3693.

02:17 And it keeps going on.

02:19 You may round that off to 1 .37 if you're going to be using a table in the book.

02:24 I'm going to use my normal cdf button.

02:27 I don't even have my table out here.

02:29 And i don't have a copy of the book.

02:31 My lower limit if i use normal cdf is that number that we just got here.

02:37 And my upper limit i'm just going to put in as like a thousand and leave the mean at zero and one for the standard deviation.

02:44 And when i do that, i get 0 .085...

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