Hypothesis Testing | Population Mean: σ Known | Lower Tail Test

The critical value approach involves determining a critical value. For a lower-tail test with known Sigma, the critical value is denoted as -Z-alpha. This is the Z-value corresponding to an area of alpha in the lower tail of the Z-distribution. With alpha = 0.05, the critical value is found by looking up 0.05 in the Z-table, which yields approximately -1.64 (or -1.65). The rejection rule for the critical value approach in a lower-tail test is to reject the null hypothesis if the test statistic is less than or equal to the critical value. Since the calculated test statistic of -2.12 is less than -1.64, the null hypothesis is rejected.

The video emphasizes that both the p-value approach and the critical value approach should always lead to the same conclusion regarding the rejection or non-rejection of the null hypothesis. If different conclusions are reached, it indicates an error in the calculations.

The video introduces hypothesis tests for the population mean when the population standard deviation (Sigma) is known. It highlights three forms of hypothesis tests: lower tail, upper tail, and two-tail, and specifies focusing on lower tail tests first. A lower-tail test is characterized by a null hypothesis where the population mean is greater than or equal to a certain value (mu naught) and an alternative hypothesis where it is less than mu naught. An example is set up with mu >= 20 (null) and mu < 20 (alternative), a sample size of 50, sample mean of 19.4, population standard deviation of 2, and an alpha level of 0.05.

The first crucial step in any hypothesis test is calculating the test statistic. For population mean tests with known Sigma, the formula for the Z-statistic is (X-bar - mu naught) / (Sigma / sqrt(n)). This statistic follows a standard normal (Z) distribution. Using the example values, the Z-statistic is calculated as (19.4 - 20) / (2 / sqrt(50)), resulting in -2.12.

The p-value approach involves calculating the p-value, which represents the probability of obtaining a test statistic as extreme as, or more extreme than, the observed value, assuming the null hypothesis is true. For a lower-tail test, it's the probability of getting a value as small as or smaller than the sample's test statistic. In this case, the p-value is the probability that Z is less than or equal to -2.12. Using a Z-table, the area to the left of -2.12 is found to be 0.0170. The rejection rule for the p-value approach is to reject the null hypothesis if the p-value is less than or equal to the significance level (alpha). Since 0.0170 is less than 0.05, the null hypothesis is rejected, concluding that the population mean is less than 20.