Testing claims about the population mean \(\mu\) when the population standard deviation \(\sigma\) is unknown

Assume that each of the samples below are a random sample from a population that is approximately normally distributed.

For each of the following problems:

1. State the claim in terms of an expression involving the population mean.
2. Determine whether the claim corresponds to a null or alternative hypothesis.
3. Determine the complementary hypothesis to the claim.
4. State the null and alternative hypotheses.
5. Determine the type of evidence, involving the sample mean, which would be evidence against the null hypothesis.
6. Determine the critical value(s) \(t_{\text{c}}\) for the rejection region(s) which will give a hypothesis test of size / significance level \(\alpha\) when the null hypothesis is true.
   • Use the \(T\)-test Demo to find the approximate critical value(s).
   • Use the table of \(t\)-values in Appendix A-3 of Triola and Triola or here to find more precise critical value(s).
7. Draw the density histogram for \(T\) under the null hypothesis, indicating the rejection region(s) and critical value(s).
8. Determine the observed value of the \(T\)-statistic \(t_{\text{obs}}\) from the sample properties.
9. Determine whether the observed \(T\)-statistic gives evidence against the null, and indicate the observed \(T\)-statistic on the density histogram from Part 7.

1. Claim: The population mean is equal to 37.

Sample Properties: \(n\) = 27, \(\bar{x}\) = 55.48, \(s\) = 24.34, \(\alpha\) = 0.01

2. Claim: The population mean is not equal to 26.

Sample Properties: \(n\) = 21, \(\bar{x}\) = 33.62, \(s\) = 18.33, \(\alpha\) = 0.1

3. Claim: The population mean is at most 77.

Sample Properties: \(n\) = 13, \(\bar{x}\) = 73.78, \(s\) = 9.56, \(\alpha\) = 0.1

4. Claim: The population mean is less than 7.

Sample Properties: \(n\) = 30, \(\bar{x}\) = 5.40, \(s\) = 6.57, \(\alpha\) = 0.1

5. Claim: The population mean is equal to 51.

Sample Properties: \(n\) = 17, \(\bar{x}\) = 53.93, \(s\) = 10.36, \(\alpha\) = 0.05

6. Claim: The population mean is greater than 1.

Sample Properties: \(n\) = 31, \(\bar{x}\) = 0.85, \(s\) = 1.05, \(\alpha\) = 0.05

7. Claim: The population mean is at most 37.

Sample Properties: \(n\) = 7, \(\bar{x}\) = 33.99, \(s\) = 10.63, \(\alpha\) = 0.05

8. Claim: The population mean is less than 81.

Sample Properties: \(n\) = 30, \(\bar{x}\) = 80.89, \(s\) = 47.96, \(\alpha\) = 0.05

9. Claim: The population mean is greater than 21.

Sample Properties: \(n\) = 31, \(\bar{x}\) = 18.85, \(s\) = 12.18, \(\alpha\) = 0.01

10. Claim: The population mean is greater than 31.

Sample Properties: \(n\) = 26, \(\bar{x}\) = 29.05, \(s\) = 16.08, \(\alpha\) = 0.1