Hypothesis Tests: Test With Unknown Variance of Population

00:00 All right, great.

00:02 Now that we know what the P value is and how to use it, we will get back to hypothesis testing. We saw only one of two possible cases.

00:11 Remember, we haven't covered the more commonly observed case when the population variance is unknown.

00:18 All right. Imagine you are the marketing analyst of a company and that you've been asked to estimate if the email open rate of one of the firm's competitors is above your company's. Your company has an open rate of 40%.

00:33 An email open rate is a measure of how many people on the email list actually open the emails they ever see.

00:40 At first, you struggle to figure out how to get such specific information about a competitor company.

00:46 But then you see that an employee of that competitor company posted a selfie on Facebook saying lol.

00:52 The email management software we are using drives me nuts.

00:57 In the background. You can see her screen, and it shows clearly the summaries of the last ten email campaigns that were sent and their corresponding open rates. Bingo.

01:09 With your statistical skills, that's all you need.

01:12 A little help from Facebook.

01:16 Let's state the hypotheses.

01:18 Null hypothesis.

01:20 Mean open rate is lower or equal to 40%.

01:24 Alternative hypothesis mean open rate is higher than 40%.

01:30 Note that in hypothesis testing, we are aiming to reject the null hypothesis when we want to test if the open rate is higher than 40%.

01:38 The null hypothesis actually states the opposite statement.

01:42 Also pay attention that this time, we are dealing with a one sided test.

01:49 All right. Your boss told you that 0.05 is an adequate significance level for this test, so that's what you'll use.

01:59 Here's the data set.

02:00 You calculate the sample mean and get 37.7%.

02:05 The sample standard deviation is 13.74%.

02:09 Thus, the standard error is 4.34%.

02:14 You assume that the population of open rates of sent emails is normally distributed? Like confidence intervals with variants, unknown and a small sample.

02:23 The correct statistic to use is the T statistic.

02:27 Remember, you do not know the variance and the sample is not big enough.

02:32 This means that the variable follows the student's T distribution, and you must employ the T statistic.

02:40 Let's calculate it. Then we calculate the T score the same way as the Z score.

02:46 The score is equal to the sample mean minus the hypothesized mean value divided by the standard error.

02:55 The result that we get is -0.53.

03:00 As we said earlier, it is easier to work with positive numbers.

03:03 So we should compare the absolute value of -0.53 with the appropriate T with n minus one degrees of freedom at 0.051 sided significance.

03:17 We quickly navigate through the table and get 1.83 at the 5% significance critical value.

03:25 Ok 0.53 is lower than 1.83.

03:30 Remember the decision rule? If the absolute value of the T score is lower than the statistic from the table, we cannot reject the null hypothesis.

03:40 Therefore, we must accept it.

03:44 What you do next is you go and tell your boss that at this level of significance, statistically, we cannot say that the email open rate of our competitors is higher than 40%. Ok What about the second measurement we saw? What was that? Ah yes.

04:02 The P value.

04:04 The p value of this statistic is 0.30 for.

04:09 As the p value is greater than the significance level of 0.05, we come to the same conclusion.

04:16 We cannot reject the null hypothesis.

04:20 Let's do a quick check.

04:23 If the significance level was 0.01, the p value would still be higher, and we wouldn't reject the null hypothesis.

04:30 This is an important observation that we haven't noted before.

04:34 If we cannot reject a test at 0.05 significance, we cannot reject it at smaller levels either.

04:42 All right. That's all for now.

04:43 Make sure you learn the material by doing the exercises after this lesson.

04:48 Thanks for watching.