Question. How to perform Hypothesis test for mean of a normal distribution with known/unknown variance?  What test statistics are used, respectively? Decision Rules? How to determine the Null and Alternative hypothesis? One-sided or two sided?Question. Ask ChatGPT to give examples.

To perform a hypothesis test for the mean of a normal distribution, we follow these steps:

1. State the null hypothesis (H0) and alternative hypothesis (H1):

   • Null hypothesis (H0): The mean of the population is equal to a specific value.
   • Alternative hypothesis (H1): The mean of the population is not equal to the specific value.

2. Determine the significance level (α) for the test. This determines the probability of rejecting the null hypothesis when it is true.

3. Choose the appropriate test statistic based on whether the population variance is known or unknown:

   • If the population variance is known, we use the z-test statistic.
   • If the population variance is unknown, we use the t-test statistic.

4. Calculate the test statistic using the sample data and the chosen test statistic formula.

5. Determine the critical region or rejection region based on the test statistic and the significance level. This is the range of values that, if the test statistic falls within, leads to rejecting the null hypothesis.

6. Compare the test statistic to the critical region. If the test statistic falls within the critical region, we reject the null hypothesis. Otherwise, we fail to reject the null hypothesis.

7. Make a decision and interpret the results in the context of the problem.

Regarding the decision rules, if the test statistic falls within the critical region, we reject the null hypothesis. If it falls outside the critical region, we fail to reject the null hypothesis.

The choice between a one-sided or two-sided test depends on the research question and the alternative hypothesis. If we are interested in determining whether the mean is significantly greater or smaller than the specific value, we use a one-sided test. If we want to determine whether the mean is significantly different from the specific value (either greater or smaller), we use a two-sided test.

Examples of hypothesis tests for mean:

1. Known variance: Suppose we want to test whether the mean weight of a certain product is 10 kg. We collect a sample of 50 products and find that the sample mean weight is 9.8 kg. We know that the population variance is 2 kg^2. Using a significance level of 0.05, we can perform a z-test to determine if the mean weight is significantly different from 10 kg.

2. Unknown variance: Let's say we want to test whether the mean height of a population is 170 cm. We collect a random sample of 30 individuals and find that the sample mean height is 168 cm. We also calculate the sample standard deviation, which is 5 cm. Using a significance level of 0.05, we can perform a t-test to determine if the mean height is significantly different from 170 cm.

Remember, the specific details and calculations may vary depending on the specific problem and data available.