Hypothesis Tests: Test With Known Variance of Population

00:01 All right. Now that we've covered the necessary theory, it is time for some testing. We're going to explore two types of tests drawn from a single population and drawn from multiple populations.

00:14 This is very similar to confidence intervals for a single population and confidence intervals for two populations that we covered previously.

00:23 In the next few videos, we will run tests for a single mean with both known variants and unknown variants.

00:31 Let's start with a test in which the variance is known, shall we? For this test, we will use our good old data scientist salary example.

00:41 Here's the data set one more time.

00:44 By now. I hope you are able to calculate the sample mean.

00:48 It is $100,200.

00:52 The population variance is known and its standard deviation is equal to 15,000. Moreover, the sample size is 30. However, you saw that according to Glassdoor, the popular salary information website, the mean data scientist salary is 113,000. The sample that is available on Glassdoor is based on self-reported numbers, and you would like to see if its value is correct.

01:21 We needed a two sided test as we are interested in knowing both of the salary is significantly less than that or significantly more than that.

01:31 The null hypothesis is the population means salary is 113,000. We denoted as MU zero equals 113,000.

01:45 The alternative hypothesis is that the population means salary is different than 113,000. All right.

01:54 Formula time, almost.

01:57 Testing is done by standardizing the variable at hand and comparing it to the lowercase c which follows a standard normal distribution.

02:06 Remember standardization.

02:08 We learned about it in the previous section.

02:10 Back then, I told you it was very important.

02:13 And you will now see why.

02:16 For those that don't remember, I suggest watching the video on standardization once again. For the others, I will quickly go through it.

02:24 We standardize a variable by subtracting the mean and dividing by the standard deviation. Since it is a sample, we use the standard error.

02:34 Thus the formula for standardization becomes.

02:39 Capital Z is equal to the sample mean minus the value of interest from the null hypothesis divided by the standard error.

02:50 In this way, we obtain a distribution with a mean of zero and a standard deviation of one.

02:57 This uppercase C should not be mistaken with lowercase c.

03:02 The Upper Casey is the standardized variable associated with the test and will be called the Z score from now on.

03:11 The lowercase c is the one from the table that we've talked about before, and henceforth will be referred to as the critical value.

03:20 All right. How does testing work? Think about this.

03:25 The lowercase c is normally distributed with a mean and standard deviation of one.

03:29 The uppercase C is normally distributed with a mean of x bar minus MU zero and a standard deviation of one.

03:38 Standardization lets us compare the means.

03:41 The closer the difference of X bar and MU 0 to 0, the closer the z score itself to zero.

03:49 This implies a higher chance to accept the null hypothesis.

03:54 Let's go back to the example.

03:56 So what is the value of our standardized variable? We plug in the numbers that we have from the beginning of the lesson.

04:04 What we get is a z-score of -4.67.

04:09 Now we will compare the absolute value of -4.67 with a lowercase z of alpha divided by two, where alpha is a significance level.

04:19 Note that we use the absolute value, as it is much easier to always compare positive capital Z's with positive lowercase c's.

04:27 Moreover, some Z tables don't include negative values.

04:31 You should be aware that the two statements -4.67 is lower than the negative. Critical value is the same as 4.6.

04:40 Seven is higher than the positive critical value.

04:44 Thus, our decision rule becomes absolute value of the z score should be higher than the absolute value of the critical value.

04:52 Using 5% significance.

04:54 Our alpha is 0.05.

04:57 Since it is a two sided test, we check the table for Z of 0.0 to 5. The corresponding value is 1.96.

05:07 The last thing we need to do is compare our standardized variable to the critical value.

05:12 If the Z score is higher than 1.96, we would reject the null hypothesis.

05:18 If it is lower, we will accept it.

05:22 4.67 is higher than 1.96.

05:25 Therefore, we reject the null hypothesis.

05:28 The answer is that at the 5% significance level, we have rejected the null hypothesis or at 5% significance.

05:36 There is no statistical evidence that the mean salary is $113,000.

05:43 There are many other ways to express this, and you will probably hear more about this later on in the course.

05:50 What if we had a different significance level? Using 1% significance.

05:54 We have an alpha of 0.01, so z of alpha divided by two is 2.58.

06:02 Once again, our z-score of 4.67 is higher than 2.58. So we would reject the null hypothesis even at the 1% significance.

06:14 But how much further can we go before we could not reject the null hypothesis anymore? 0.5%. 0.1%.

06:22 There is a special technique that allows us to see what the significance level is, after which we will be unable to reject the null hypothesis.

06:30 We will see it in our next video.

06:33 Stay tuned.