Confidence Intervals for Dependent Samples of Two Means

00:00 So far, we have talked about confidence intervals with population variances that are either known or unknown.

00:07 However, we were considering only one population.

00:11 In the next couple of lessons, we will explore confidence intervals looking into two populations. These cases are more important as they have a wide range of real world applications.

00:23 A few important distinctions need to be made before we dive into this topic.

00:28 In some cases, the samples that we have taken from the two populations will be dependent on each other, and in others they will be independent.

00:36 Dependent samples are easier.

00:38 You will experience this firsthand.

00:40 Dependent samples can occur in several situations.

00:45 First, when we are researching the same subject over time.

00:49 Examples are weight loss and blood samples.

00:52 Essentially, we are looking at the same person before and after.

00:57 These two examples will be explored and detailed in these lessons.

01:02 Another case in which we have dependent samples is when investigating couples or families, for instance, habits of husbands and wives.

01:11 They are obviously dependent on each other, as the time these people spend together at home often coincides.

01:17 Watching TV, eating dinner, often sharing the same household income.

01:24 Finally, we can have the same people, but in samples relating to different things.

01:28 So instead of a before and after situation, we are looking at cause and effect.

01:34 For example, when applying to university in the US you sit the SAT and based on it, you either get admitted or you don't.

01:42 The applicant is the same person.

01:44 However, the samples are different.

01:46 One relates to the SAT.

01:48 The other to the admittance outcome.

01:52 In terms of testing, we have one formula for confidence intervals, for dependent samples and other statistical methods like regressions, which we will study later on.

02:02 For now, let's stick with confidence intervals.

02:05 Ok. When we have independent samples, we can further distinguish three cases when the population variance is known, when the population variance is unknown but assumed to be equal, and when it is unknown and assumed to be different.

02:22 Sounds a bit overwhelming, but don't worry.

02:25 In statistics, many concepts are similar to each other, and you will quickly see that you have already acquired the intuition that allows you to understand these concepts pretty fast.

02:37 All right. Let's get on to the topic of this lesson.

02:41 The dependent samples.

02:44 This statistical test is often used when developing medicine.

02:48 Let's say you have developed a pill that increases the concentration of magnesium in the blood. It is very promising, but there is no data to support your claim.

02:58 After testing the drug in a laboratory, it is time to see its actual effect on people. What you would typically do is take a sample of ten people and test their magnesium levels before and after taking the pill.

03:11 The two dependent samples are the magnesium levels before and the magnesium levels after.

03:17 It is clear that it is the same people we are testing.

03:20 Thus, the samples are dependent.

03:23 An important note is that the populations are normally distributed.

03:27 Actually, when dealing with biology, normality is so often observed that we immediately assume that such variables are normally distributed.

03:36 Okay. Back to the example.

03:39 Whenever you take a blood test, the magnesium levels are stated in milligrams per deciliter, and a healthy person would usually have somewhere between 1.7 and 2.2 milligrams of magnesium per deciliter.

03:52 Here is a table that contains a sample of ten people and their levels of magnesium before and after taking the pill for some time.

04:00 We've also added a cell that calculates the difference in levels before and after taking the pill. Instead of dealing with two variables, we now have only one. In this way, the data looks as a single population, doesn't it? Let's calculate the mean and the standard deviation of the differences.

04:19 The mean is 0.33 and the standard deviation is 0.45.

04:24 Moreover, we know that the sample size is ten.

04:28 The formula that would allow us to calculate a confidence interval is the following.

04:34 The population is normally distributed, but the sample we have contains only ten observations. Therefore, the distribution will have to work with is students T and the appropriate statistic is T.

04:47 You can clearly see that it is the same as the one for a single population with an unknown variance.

04:53 Let's choose the level of confidence and plug in the numbers.

04:57 As we have said many times.

04:58 95% confidence is one of the most common levels.

05:02 And so we will use it here as well.

05:05 The T statistic with nine degrees of freedom for a 95% confidence interval is 2.26. Now we have everything we need, and we can calculate the confidence interval.

05:18 It lies in the range between 0.01 and 0.65.

05:23 How do we interpret this result? Well, in 95% of the cases, the true mean will fall in this interval. Moreover, the whole interval is positive.

05:34 This shows that the true mean of the difference is definitely positive.

05:38 Therefore, with 95% certainty, we can say that the levels of magnesium in the test subject's blood is higher.

05:46 The purpose of this test was to determine whether the drug is effective.

05:50 Based on our small sample, it most likely is.

05:54 All right. This shows you some of the practical applications of inference.

05:59 Stay tuned for our next lesson, in which we will explore confidence intervals on independent samples.