Question

Suppose we want to test H0: μ = 50 versus Ha: μ> 50, where μ is the average of a Normal random variable with a standard deviation equal to 10. Once a sample of n = 36 population elements was extracted, x(mean) = 53 was observed . Test using the levels of 1%, 2%, 5% and 10%.

Answer 1

To test the hypothesis H0: μ = 50 versus Ha: μ > 50 using a one-sample z-test with a significance level of 1%, 2%, 5%, and 10%, the test statistic is calculated to be 1.8. Since the test statistic is not greater than the critical values for any of the given significance levels, we fail to reject the null hypothesis. Therefore, there is not enough evidence to support the claim that the population mean is greater than 50 at any of the given significance levels.

Step-by-step explanation:

To test the hypothesis H0: μ = 50 versus Ha: μ > 50, where μ is the average of a Normal random variable with a standard deviation of 10, we can use a one-sample z-test.

Step 1: Formulate the hypotheses:

• Null hypothesis (H0): The population mean (μ) is 50.
• Alternative hypothesis (Ha): The population mean (μ) is greater than 50.

Step 2: Set the significance level (α) and determine the critical value(s) based on the desired confidence level:

• For a 1% significance level, the critical value is z = 2.326.
• For a 2% significance level, the critical value is z = 2.054.
• For a 5% significance level, the critical value is z = 1.645.
• For a 10% significance level, the critical value is z = 1.282.

Step 3: Calculate the test statistic:

• The test statistic (z) can be calculated using the formula: z = (x - μ) / (σ / sqrt(n)), where x is the sample mean, μ is the hypothesized population mean, σ is the population standard deviation, and n is the sample size.
• In this case, x = 53, μ = 50, σ = 10, and n = 36.
• Calculating the test statistic gives z = (53 - 50) / (10 / sqrt(36)) = 1.8.

Step 4: Compare the test statistic with the critical value(s) and make a decision:

• If the test statistic is greater than the critical value, reject the null hypothesis. If the test statistic is less than or equal to the critical value, fail to reject the null hypothesis.
• In this case, since the test statistic (z = 1.8) is not greater than the critical value for any of the given significance levels, we fail to reject the null hypothesis at all significance levels.

Therefore, there is not enough evidence to support the claim that the population mean is greater than 50 at any of the given significance levels.