Confidence Intervals for Independent Samples With Unknown but Assumed Population Variance

00:01 Hi again. In the last few lessons, we've been focusing on confidence intervals.

00:06 We'll do that here too.

00:08 This lesson is about independent samples with variances unknown but assumed to be equal. All right.

00:16 Think about this example.

00:18 You are trying to calculate the difference of the price of apples in New York and LA.

00:24 You go to ten grocery shops in New York, and your friend Paul, who lives in L.A., visits eight grocery shops in order to help you with the research.

00:33 Once you've organized the data and a table, you start reflecting on how you can create a confidence interval that shows the difference between the price of apples in New York and LA. You don't know what the population variance of apples in New York or LA is, but you assume it should be the same for NY and LA.

00:53 So you calculate the mean price in NY and LA and obtain $3.94 and $3.25 respectively.

01:03 Moreover, their sample standard deviations are $0.18 and $0.27. What should we do now? Well, we assume that the population variances are equal, so we have to estimate them. The unbiased estimate or in this case is called the pooled sample variance, and we could use the following formula to calculate it.

01:29 As you can see, it is based solely on the sample sizes and the sample standard deviations of the two data sets.

01:36 We quickly plug in the numbers and get a pooled sample variance of 0.05 and a pooled standard deviation of $0.22.

01:48 How about the statistic needed to form a confidence interval? Well, we have an unknown variance, so you guessed it.

01:57 It's the T statistic.

02:00 Here's the formula for the confidence interval.

02:05 Let's compare it to the formula for independent samples with known variants.

02:12 We have the same difference of sample means, but the variance is instead of the population variances, we have the pooled sample variance.

02:21 And then instead of the Z statistic, we have the T statistic.

02:27 Although the formulas are different, they are very consistent.

02:31 Right. You must be wondering about the PT statistic though. It is a bit stranger this time, so let's quickly clarify it.

02:41 The degrees of freedom are equal to the total sample size, minus the number of variables. So far we have seen t's with n minus one degrees of freedom because we had a sample size of NW and only one variable.

02:56 This time we have two sample sizes one of ten and one of eight observations and two variables.

03:03 Apple prices in New York and Apple Prices in LA.

03:07 Therefore, we have 16 degrees of freedom.

03:12 All right. Checking the statistic from the table.

03:16 For a 95% confidence interval with 16 degrees of freedom, we get 2.1 to.

03:23 Let's plug everything in and get our answer.

03:27 The 95% confidence interval is between 0.47 and 0.92. What's the interpretation? Well, we are 95% confident that the actual difference between the two populations price of apples in New York and in L.A.

03:43 is somewhere between 0.47 and 0.92.

03:49 Therefore, it is clear that apples in New York are much more expensive than in L.A.. Good job and thanks for watching.