Hypothesis Tests: Independent Samples With Unknown Assumed Variance

00:00 This is the final lesson we will do on testing.

00:04 The last case we'll examine here is the one with independent samples and unknown variances, which are assumed to be equal.

00:12 I'll quickly brush up your memory on the data set we did and the confidence interval section. You were trying to see if apples in New York are more expensive than the ones in LA.

00:23 You went to ten grocery shops in New York, and your friend Paul, who lives in LA, went to eight grocery shops there.

00:31 You got all the prices and put them in a table.

00:34 And what the population variance of apple prices is.

00:37 But you assume it should be the same for New York and L.A.

00:42 Let's state the null and alternative hypothesis.

00:46 Eight zero mu in New York is equal to mu in LA or New York minus mu in LA is equal to zero.

00:57 H one mu in New York is different than mu in LA Mu in New York minus moo in LA differs from zero.

01:07 All right. That's our data set.

01:10 We have also calculated the sample means standard deviations and sample sizes. What can we do when the variance is unknown but assume to be equal? Earlier. We use the pooled variance formula.

01:24 Well, here it is again, remember? All right. It's all about plugging in numbers, so I'll save you the trouble.

01:36 The pooled variance is 0.05.

01:40 One last thing we need is the standard error of the difference of means.

01:44 It is given by the following formula.

01:50 I'm going faster than usual, as we've seen all of this before.

01:53 Moreover, testing is about understanding.

01:56 Computation is routine.

01:58 So let's start testing, shall we? Small samples, variance unknown.

02:04 Which statistic do we need? Exactly. It's the T statistic again.

02:10 How many degrees of freedom? You may recall it from earlier, it was the combined sample size minus the number of variables. So ten plus eight minus two, which gives us 16. Let's see the t statistic formula. Once again, the difference between sample means minus the difference between hypothesized true means divided by the standard error.

02:38 After plugging in everything, we get a test statistic of 6.53.

02:45 Do we need to compare it? This is by far the most extreme test statistic we have seen.

02:51 You will have a hard time finding it in the tea table.

02:55 For common tests. A rule of thumb is to reject the null hypothesis when the RT score is bigger than two.

03:02 Generally, for Z score and T score, a value that is higher than four is extremely significant. Let's see the two sided p value.

03:13 The P value of this test is lower than 0.000.

03:17 Somewhere around 0.000001.

03:22 In our lesson about P value, we said that researchers are always looking for those three zeroes after the dot.

03:28 It means that the test is extremely significant and the probability of making a type one error is virtually zero.

03:36 Therefore, we reject the null hypothesis at all common and uncommon levels of significance. There is a strong statistical evidence that the price of apples in New York differs from in LA.

03:50 But such an extreme result may also mean that the hypothesis is pointless or poorly designed. From the mean values of 3.94 and 3.25, and with such small and close standard deviations of around 0.2, we could easily say that the prices are different.

04:06 No testing needed.

04:08 A much more interesting question would be if the price of apples in New York is 20% higher than that in LA.

04:16 I will leave you this exercise for homework.

04:19 All right. We are done with hypothesis testing.

04:23 Cheers. And thanks for watching.