Hypothesis Testing: Dependent Samples

00:00 We are not done with hypothesis testing just yet.

00:05 Remember how we started with confidence intervals for a single population mean and then switch to examples considering multiple populations.

00:12 Well, we are in the same situation here.

00:15 Single populations are just the beginning.

00:19 Time to do multiple means.

00:22 We will start with dependent samples.

00:24 The most intuitive examples of dependent samples are the ones you have been through, like weight loss and blood tests.

00:32 The sample is drawn from weight loss data or concentration of nutrients data.

00:36 But the subject of interest is the same person before and after.

00:43 Okay. Let's get to work.

00:45 Do you recall our example with the magnesium levels in one's blood? There was this drug company developing a new pill that supposedly increased levels of magnesium recipients.

00:56 There were ten people involved in the study that were taking the drug for some time, and we calculated confidence intervals that helped us study the effects of that drug.

01:05 They indicated the range of plausible values for the population mean.

01:11 This time, we want to come to a single, definite conclusion about the effectiveness of the drug. All right, let's state the null hypothesis.

01:22 The population mean before is greater or equal than the population mean after.

01:28 The alternative is that the population mean before is lower than the one after.

01:34 Once again, we want to know if the magnesium levels are higher.

01:39 We construct the null and alternative hypotheses in such a way so that we are aiming to reject the null hypothesis.

01:46 We expect the levels to be higher, so when the null hypothesis, we state them to be lower or equal.

01:53 Okay. Let's re-order a bit.

01:57 H zero is a mu before which is equal or higher than a mu after.

02:02 This is equivalent to u before minus mu after is equal to zero or positive. We can substitute this with Capital D zero.

02:12 It stands for the hypothesized population mean difference.

02:16 So we restate our hypotheses using D for simplicity.

02:21 Now we have our test designed.

02:23 Let's crunch some numbers.

02:26 Here's the data set.

02:29 We have ten observations.

02:30 People have registered before and after.

02:33 Naturally, the difference is equal to before, minus after.

02:37 We can calculate the sample mean of the difference.

02:41 We get -0.33.

02:44 The sample standard deviation is 0.45 and the standard error is 0.14. The appropriate statistic to use here is the t statistic.

02:56 We have a small sample.

02:57 We assume normal distribution of the population, and we don't know the variance.

03:02 So the T score is equal to the following expression.

03:08 Now we can simply carry out the calculations and find that its value is -2.29. Since we don't want to choose a level of significance, let's solve this problem with the p value.

03:22 In order to find the p value of this one sided test, we may go to the table and see it as somewhere between 0.01 and 0.025.

03:32 As I told you earlier, using software is much easier.

03:36 So after using an online p value calculator, I can tell you that it is exactly 0.024.

03:46 What was the decision rule again? If the p value is lower than the significance level we are interested in, we reject the null hypothesis.

03:56 Okay. So if the level of significance is 0.05 and the P value is lower, we will be able to reject the hypothesis at 5%.

04:08 If the level of significance is 0.01, however, the p value is higher, so we cannot reject the null hypothesis at a 1% level of significance. The lowest value for which we can reject it is 0.024, which is exactly the p value.

04:27 What does this tell us? Well, it is up to the researcher to choose the level of significance in the case of the magnesium pill.

04:35 We expect that the researcher will be very cautious, as he would want to know if this is an effective pill that will be able to actually help people.

04:43 If we cannot say that the pill works at a 1% significance level, perhaps it is better to take it back to the laboratory.

04:50 An alternative would be to test again and increase the sample size for better results.

04:56 A sample of 100 people would improve the level of precision significantly.

05:02 All right. So we've done some more hypothesis testing, and we've explored some factors that help you determine the significance level of the test.

05:11 Stay with us for our next lesson, where we will learn how to test independent samples.

05:17 Thanks for watching.