Summary
Highlights
This module introduces hypothesis testing for the population mean. The objectives are to distinguish between population mean and sample mean comparisons and solve problems involving hypothesis testing for the population mean.
There are three cases in hypothesis testing for the population mean: when the population standard deviation is known, when it is unknown, and when it is unknown but the sample size exceeds 30. The video provides a formula for the test statistic when the standard deviation is known.
A problem is presented: a random sample of 100 deaths in the Philippines showed an average lifespan of 71.8 years with a standard deviation of 8.9 years. The question is whether this indicates the average lifespan today is greater than 70 years, using a 0.05 level of significance.
The first step is to state the null (Ho) and alternative (Ha) hypotheses. For the given problem, the alternative hypothesis (Ha) is that the average lifespan is greater than 70 years, indicating a significant relationship. The null hypothesis (Ho) states the average lifespan is equal to 70 years.
The level of significance (alpha) is 0.05, as given in the problem. The third step is to determine the critical region. Using a Z-table for a one-tailed test with alpha = 0.05, the critical value of Z is 1.645.
The Z-value is computed using the formula: Z = (sample mean - initial mean) / (standard deviation / sqrt(n)). Given the sample mean (71.8), initial mean (70), standard deviation (8.9), and sample size (100), the calculated Z-value is 2.02.
The calculated Z-value (2.02) is compared to the critical value (1.645). Since 2.02 is greater than 1.645, the null hypothesis (Ho) is rejected. The conclusion is that the average lifespan today is indeed greater than 70 years, based on the significant relationship found.