Confidence Intervals for Independent Samples and Known Variance of the Population

00:00 In our last lesson, we learned about confidence intervals for two means based on dependent samples.

00:06 In this video, we will explore independent samples.

00:10 As we said earlier, there are three subcases known population variances, unknown population variances, but assumed to be equal and unknown population variances that are assumed to be different.

00:24 Here, we will focus on independent samples with known population variances.

00:30 All right. Let's jump right into the example that would allow us to understand the concept a bit better.

00:37 We would like to test the grades of two departments in a UK university.

00:41 University grades in the UK expressed in percentages.

00:45 Our samples are taken from two departments, engineering and management.

00:50 The table you see here summarizes the data.

00:54 We have a sample of 100 engineering students and the average grade is 58%.

01:00 From past years, we know that the population standard deviation is ten percentage points.

01:05 Thus, the variance is known.

01:08 We also have a sample of 70 students from the management department, and we've got a sample mean of 65%.

01:15 Again, from past data, we know that the population standard deviation is five.

01:22 Okay. But before we get on with calculations, let's point out three important considerations. First, the populations are normally distributed. That's a fair assumption as we are dealing with grades.

01:35 Second, the population variances are known.

01:39 And third, the sample sizes are different.

01:42 Different sample sizes are a common occurrence in the real world.

01:46 In our previous lesson, when we had dependent samples, we were testing the same people over time, so it made sense that the sample sizes were equally big.

01:55 This time, however, the observations are completely different.

01:59 They are students from different departments with different teachers obtaining different grades when taking different exams.

02:06 The grade of a person from engineering doesn't in any way affect the grade of a person studying management.

02:12 Right. The two samples are truly independent.

02:17 All right. What are we testing? We want to find a 95% confidence interval for the difference between the grades of the students from engineering and management.

02:27 As with every confidence interval, we must identify the test statistic.

02:32 Samples are big, population variances are known and populations are assumed to follow the normal distribution.

02:39 All this information points us to the Z statistic instead of the T.

02:45 The last ingredient is the variance.

02:47 We haven't been through enough mathematics in this course to derive the formula, so we will simply state it.

02:53 The variance of the difference between the two means is equal to the variance of the grades received by engineering students divided by the sample size of engineering students, plus the variance of grades obtained by management students divided by the sample size of management students.

03:10 The underlying logic is that dispersion is additive.

03:14 More variables means higher or equal variability.

03:19 Okay. So what is the confidence interval formula this time? Well, it is given by the expression below.

03:27 X bar minus y bar is the difference point estimate.

03:31 Z is a statistic.

03:35 We plug in the numbers and get a confidence interval between -9.28 and -4.72. All right.

03:44 What's the interpretation? We are 95% confident that the true mean difference between engineering and management grades falls into this interval.

03:53 Know that this time the whole interval is negative.

03:57 This is because we were calculating engineering grade minus management grade.

04:01 As the engineers were consistently getting lower grades, the difference is negative.

04:07 Had we calculated the difference is management minus engineering, we would have obtained a confidence interval between 4.72 and 9.28, completely symmetrical around zero.

04:19 All right. Great.

04:21 Remember to do all the exercises available in the course materials, as the topics are getting tougher and tougher.

04:28 See you next time, and thanks for watching.