Hypothesis Testing: Independent Samples and Known Variance

00:00 Hi again.

00:02 As you probably expected in this lesson, we will learn about independent samples with known variants.

00:08 Let's get into the example right away.

00:11 You may remember this one.

00:12 We are about to test the average grades of students from two different departments in a UK university.

00:18 I would like to remind you that in the UK grades are expressed in percentages.

00:24 The two departments are engineering and management.

00:28 We were told by the dean that engineering is a tougher discipline and people tend to get lower grades. He believes that on average, management students outperform engineering students by four percentage points.

00:42 Now it is our job to verify if that is the case.

00:47 Let's state the two hypotheses.

00:50 H zero is the difference between the means of the two populations is fine.

00:55 As for. By the way, notice that I can make h zero engineering minus management and get a negative difference.

01:05 Or I can make h zero management minus engineer and get a positive difference. Either way works.

01:13 Just so we can see as many different situations as possible.

01:16 I will keep the difference negative.

01:20 So h one is the population mean difference is different than for.

01:27 Once again, this is a two sided test.

01:29 Our research question is not to find the difference, but to check if it is exactly for. Right.

01:37 Let's get our hands dirty.

01:39 Here's the table that summarizes the data.

01:43 The sample sizes are 170 respectively.

01:48 The sample means are 58% and 65%, and the population standard deviations are 10% and 6% and are known from past data.

01:59 If you remember, when the population is known for independent samples, the standard error of the difference is equal to the square root of the sum of the variance of engineering divided by its sample size and the variance of management again divided by its sample size.

02:16 All we have left is to compute the test statistic.

02:19 We have big samples and known variances.

02:22 Therefore, we can use the Z statistic.

02:26 I hope you are getting the point.

02:27 Small samples and unknown variances means T large sample and known variances mean z.

02:35 When we have large samples and unknown variances, it is up to the researcher, but generally it is okay to use Z in that case as well.

02:44 All right. Here's the formula for the test statistic.

02:49 Sample difference mean minus hypothesized difference mean divided by the standard error. We plug in the numbers and get a Z score of -2.4 for.

03:01 Let's calculate the p value.

03:03 Once again, I'll just tell you the p value.

03:05 As usually, you will obtain it using a software.

03:09 The P value of the two sided test is 0.015.

03:15 What we can say is that at 5% significance, which is common for such a study, the p value of 0.015 is lower than 0.05.

03:24 Thus we reject the null hypothesis.

03:27 There is enough statistical evidence that the difference of the two means is not 4%.

03:33 All right, cool.

03:34 Here's a trick.

03:36 What if you want to know if the difference is higher or lower than for.

03:40 The sign of the test statistic can give you that information.

03:44 A minus sign of the test statistic means it's smaller.

03:48 If you reverse engineer the standardization process, you will find that true value is likely to be lower than the hypothesized value.

03:56 In our case, this translates into the true mean is likely to be lower than minus four. Lower than minus four entails that possible values are minus five, minus six, and so on.

04:09 This is additional information that you can give to the dean.

04:13 All right. Done with that lesson, too.

04:16 Let's proceed to the final topic.

04:18 Independent samples and unknown variances.