The owner of a local chain of grocery stores is always...

The owner of a local chain of grocery stores is always trying to minimize the time it takes her customers to check out. In the past, she has conducted many studies of the checkout times, and they have displayed a normal distribution with a mean time of 12 minutes and a standard deviation of 2.3 minutes. She has implemented a new schedule for cashiers in hopes of reducing the mean checkout time. A random sample of 28 customers visiting her store this week resulted in a mean of 10.9 minutes. Does she have sufficient evidence to claim the mean checkout time this week was less than 12 minutes? Use $\alpha=0.02$

Key Concepts

Hypothesis Testing

Hypothesis testing is a statistical method used to decide whether there is enough evidence in a sample to infer that a certain condition holds for the entire population. In this problem, the hypothesis test is used to determine if the new cashier schedule has effectively reduced the mean checkout time from the historical value.

Null and Alternative Hypotheses

The null hypothesis (H?) represents the status quo or the claim that no change has occurred, while the alternative hypothesis (H?) represents the claim that there has been a change. For this problem, the null hypothesis is that the mean checkout time is 12 minutes, and the alternative hypothesis is that it is less than 12 minutes, reflecting a one?sided test.

Significance Level

The significance level, denoted by alpha (?), is the threshold for rejecting the null hypothesis. It represents the acceptable probability of committing a Type I error, which is rejecting a true null hypothesis. In this scenario, the significance level is set at 0.02, meaning there is a 2% risk of incorrectly rejecting the null hypothesis.

Test Statistic

The test statistic is a standardized value that is calculated from sample data during a hypothesis test. It is used to determine how far the sample statistic lies from the null hypothesis value, in units of the standard error. This problem typically involves computing a z-test or t-test (given the sample size and knowledge about the distribution) to compare the sample mean to the historical mean.

p-value

The p-value is the probability of observing a test statistic as extreme as, or more extreme than, the one computed from the sample data, assuming the null hypothesis is true. In this context, a low p-value (lower than the significance level) would lead to rejecting the null hypothesis, thereby supporting the claim that the new schedule has reduced the mean checkout time.

Type I and Type II Errors

Type I error is the error of rejecting the null hypothesis when it is actually true, while Type II error is the error of failing to reject the null hypothesis when it is actually false. The significance level (0.02) controls the probability of a Type I error, which is especially important in contexts where falsely claiming an improvement could have significant consequences.

Transcript

0:00 Hey everyone.

00:01 So today we've been given a scenario which is looking at the time it takes for customers to check out at a grocery store.

00:07 And what's happened here is we have a manager who's trying to reduce that number.

00:11 And so she's tried a new schedule for the people who scan the items to try and reduce this time.

00:17 And so what we've been asked to do is conduct a test to see if there's sufficient evidence to claim that the mean checkout time is less than it originally was.

00:27 And so the values that we're given in the question, which, we're we're told that the normal, sorry, that the checkout times before this change displayed a normal distribution with a mean time of 12 minutes.

00:42 So i'm going to write that mu equals 12.

00:46 Because it's telling us that we've got a normal distribution and that the mean was 12, so we can write that like this.

00:53 We're also told it had a standard deviation of 2 .3 minutes.

00:59 This new schedule, having tried that for a bit, they did a sample with 28 customers who visited the store, and they found that the average time it took them to checkout was 10 .9 minutes rather than 12.

01:13 So we're going to see if there's sufficient evidence to suggest that this has dropped.

01:18 And so to start with, let's make some hypotheses.

01:22 So an all hypothesis is that nothing has changed.

01:25 The average time to checkout is still 12 minutes...

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