13.3.13-T A quality-control manager at an amusement park...

13.3.13-T A quality-control manager at an amusement park feels that the amount of time that people spend waiting in line for the roller coaster is too long. To determine if a new loading/unloading procedure is effective in reducing wait time in line, he measures the amount of time (in minutes) people are waiting in line for seven days. After implementing the new procedure, he again measures the amount of time (in minutes) people are waiting in line for seven days and obtains the data in the accompanying table. To make a reasonable comparison, he chooses days when weather conditions are alike. Complete parts (a) through (c) below. Click the icon to view the data table. Waiting Time Day Mon Tues Wed Thurs Fri Sat Sun Wait time before new procedure 15.9 25.9 24.4 39.5 57.3 32.1 81.8 Wait time after new procedure 10.7 28.3 19.2 35.9 61.3 31.8 75.3 What is the alternative hypothesis? A. H1: ?1 = ?2 B. H1: ?1 ? ?2 C. H1: ?1 > ?2 Use technology, what is the P-value of the test? (Round to three decimal places as needed.)

Transcript

00:01 We would like to perform a hypothesis test.

00:03 We want to know if the amount of time people are waiting in line is different for a new procedure.

00:13 So specifically lower.

00:15 Okay, so let me state the null and alternative hypotheses.

00:19 The null hypothesis represents no change, no difference, nothing special happening.

00:24 So that would be that mu 1 is equal to mu 2.

00:27 Average time before, equal to average time after.

00:30 The alternative, it looks like we're going for different, so it could be higher, it could be lower, we just want to know if it's any different, and you want the p -value of this test.

00:42 Okay, so to perform a hypothesis test, you start by assuming the null hypothesis is true, that the means of the populations are the same.

00:51 And then you consider what would happen if you were to take a sample size 7 from each and look at the difference in sample means.

00:58 Now this uses the central limit theorem.

01:03 As sample size increases, sample means become more and more normally distributed.

01:08 We have really small samples here, so we have to make an assumption.

01:11 We have to assume the populations are normal, which i don't feel very good about, but that's all they've given us.

01:18 We have to do that.

01:20 Then the difference between the two normal distributions is also going to be normal.

01:26 Now, i'm honestly a little bit suspicious of their entire method for this test, because it it looks like the day has quite an impact on waiting time.

01:35 So i would prefer if they did a more complicated test that took that into account rather just comparing the means of the two weeks.

01:43 But let's do what they wanted to do.

01:47 The difference in sample means follows a normal curve.

01:51 Its mean is the difference in population means, and if we assume the null hypothesis is true, they're equal, so the difference is zero.

01:58 The standard deviation is the pooled standard error, which looks like this.

02:06 It's just the variance of the two separate.

02:08 Sampling distributions added together square root to get the standard deviation.

02:13 So what do we? well, we don't know the population standard deviations.

02:19 Therefore, we need to use a t test.

02:22 If we did have sigma 1 and sigma 2, we'd be using z instead, but we have to use t here.

02:28 Then i would like to know the p value.

02:31 So the p value is for probability of our result for something more extreme, given the null hypothesis is true.

02:43 So this curve is assuming the null hypothesis is true.

02:46 Our difference is going to be somewhere out here.

02:50 I'm going to find the probability of something that different.

02:53 Note this is a two -tail test because the alternative is just it is different.

02:58 It could be higher, it could be lower.

03:00 So we're including both tails here.

03:01 A two -tailed p value is twice for size of a be one tail of p value.

03:05 It's the difference between the probability something is this different, and the probability something is specifically this high or this low.

03:13 So we're going to have both tails.

03:15 Now, t distribution needs something with software built into it for that.

03:19 It could be your calculator, could be something like excel, whatever you like.

03:22 But you do need the test statistic...

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