Test the ( null) hypothesis, at 5% level of significance, that the population mean age of students is 45 years. Show your work.
To test the null hypothesis that the population mean age of students is 45 years at a 5% level of significance, you would conduct a hypothesis test using the following steps:
Step 1: State the hypotheses.
- Null hypothesis (H0): The population mean age of students is 45 years.
- Alternative hypothesis (Ha): The population mean age of students is not equal to 45 years.
Step 2: Set the significance level (α).
In this case, the significance level is given as 5% or 0.05.
Step 3: Collect and analyze the data.
Collect a sample of data that represents the population of students' ages. Calculate the sample mean and standard deviation of the ages.
Step 4: Calculate the test statistic.
Calculate the test statistic based on the sample data. In a hypothesis test for the mean, you typically use the t-statistic when the population standard deviation is unknown. The formula for the t-statistic is:
t = (x̄ - μ) / (s / √n)
Where:
x̄ is the sample mean
μ is the assumed population mean under the null hypothesis (45 years)
s is the sample standard deviation
n is the sample size
Step 5: Determine the critical value(s).
Look up the critical value(s) in the t-distribution table or use statistical software based on the significance level and degrees of freedom. The degrees of freedom for this test are equal to the sample size minus 1.
Step 6: Calculate the p-value.
Calculate the p-value associated with the test statistic using the t-distribution. The p-value represents the probability of obtaining a test statistic at least as extreme as the one observed, assuming the null hypothesis is true.
Step 7: Make a decision.
If the p-value is less than the significance level (α), reject the null hypothesis. Otherwise, if the p-value is greater than or equal to the significance level, fail to reject the null hypothesis.
Step 8: Interpret the results.
Based on your decision in Step 7, interpret the results in the context of the problem. If you reject the null hypothesis, you would conclude that there is sufficient evidence to suggest that the population mean age of students is not 45 years. If you fail to reject the null hypothesis, you would conclude that there is not enough evidence to suggest that the population mean age of students is different from 45 years.
Note: To completely solve the problem, specific numerical values such as the sample mean, sample standard deviation, degrees of freedom, critical value(s), t-statistic, and p-value are needed.