Hypothesis Tests: Practical Example

00:00 Hello and welcome to our third practical example.

00:04 This time, we will explore the topic of gender pay gap.

00:09 We will test whether a particular company is discriminating against some of its employees on a gender basis.

00:16 Our fictitious company is called Spark Fortress, Inc.

00:20 It is a big company with more than 5000 employees.

00:24 And here we will work with a sample of 174 of them.

00:29 We have data showing us their name, age, gender, nationality, ethnicity, tenure, department position and annual salary. I believe there is no need for further explanation of the data set.

00:45 We are going to test if there is significant difference in the salaries employees are paid based on their gender.

00:52 It would be easier if we look at the problem at hand and the following way our 174 employee sample could be divided into two sub samples, one that is exclusively male and one female.

01:06 So we have two samples drawn from the same population that are independent, although so far we've worked with different populations only if the values in one sample reveal no information about the other sample, then they are considered independent.

01:23 There are different methodologies to conduct such a study, and while regression analysis is my preferred one here, we will use a hypothesis test for the mean salary to determine if there is reasonable evidence for gender discrimination.

01:37 Let's state the two hypotheses.

01:41 Eight zero. The average male salary is equal to the average female salary, or mu m minus mu f equals zero. H1.

01:53 The average male salary differs from the average female salary.

01:59 Ok. The test we should use is the T test for independent samples.

02:05 What about the salary population variance? It is truly unknown, and we can assume it is equal.

02:13 Let's construct a frequency distribution table.

02:18 We have 98 females and 76 males.

02:23 These are our sample sizes.

02:26 Next, we should calculate the means and the sample variances of the two samples that we got. As we assume that the population variances are equal, we should also compute the pooled variance.

02:39 Here's the good old formula.

02:44 And here is the ginormous result.

02:49 Finally, the t score for this test is computed following the familiar expression.

02:56 We get a t score of 1.3 for.

03:00 The degrees of freedom are 172.

03:05 As we said earlier, once we have surpassed 50 degrees of freedom, the students T distribution almost completely overlaps with the normal distribution.

03:14 Thus, the P values for a T score of 1.34 and a z score of 1.34 will be virtually the same.

03:23 You already know how to use a p value calculator, so I'll just give you the p value. It's 0.182.

03:34 The P value is much greater than all common levels of significance.

03:39 We conclude that we cannot reject the null hypothesis.

03:43 There isn't enough statistical evidence that there is a wage gap in this firm.

03:48 Now. That's cool.

03:49 Spark Fortress seems like a nice place to work at, but let's dig just a bit deeper into this result.

03:58 Personally, I'm interested to know if there is no wage gap at all, or maybe there is one hidden beneath the aggregate values we just investigated.

04:07 Sometimes it is a good idea to examine the data set manually, and that's something we didn't do in the beginning, but we should have done.

04:16 Let's order the salaries from largest to the smallest.

04:22 We can see that the highest paid employee is actually the president and CEO of the company, Caroline Bold, who is female.

04:31 This may explain the egalitarian culture of the company, but it may also mean that our high salary biased our data.

04:40 What if, on average, it seems that women are rewarded the same as men, but in fact, very few of them are.

04:48 In such cases, I would normally further segment the data.

04:54 Let's divide the employees into two more groups, below 35 and above 35.

05:00 This will give us valuable information about the wage equality of younger and older staff. I've created two more data sets that are based on the original one.

05:11 Let's run the same tests as before, but this time we will do it in our segmented data. The hypotheses are the same.

05:20 What we get for these two tests is a T score of 0.43 for employees below 35 and 2.00 for employees over 35.

05:31 The corresponding P values are 0.668 and 0.048.

05:39 What these numbers mean is that in the group below 35, there is virtually no wage gap on a gender basis.

05:45 The p value is so big that we may be 100% sure there is no discrimination going on.

05:53 In the older group, however, the p value is 0.048.

05:59 This is very close to 0.05, but still below it.

06:04 This implies that at 95% significance, we reject the null hypothesis.

06:08 Therefore, a wage gap does exist for older employees.

06:15 All right. This was a two sided test, so we are not sure who gets more money. Right? Well, do you remember the nifty trick? The T score of two is positive.

06:27 Therefore, the difference in pay is positive in favor of males.

06:34 A limitation of this analysis is that we omitted other factors such as position and ethnicity. So we are not completely sure what's going on in the firm, but we can say that overall there is no wage gap in Spark Fortress and this is driven by wage equality among young employees.

06:51 This is a good indicator, as it means that the company is firmly moving towards complete equality.

06:59 All right. Your homework is to conduct a similar test that aims to capture if there is racial discrimination in the firm.

07:06 You can find it in the data, in the resources for this lesson.

07:10 Good luck, and thanks for watching.