Let's go through the quiz step by step, based on the scenario given:

1. Null and Alternate Hypotheses:

   • Null Hypothesis (H₀): The mean height of adult males in this population is 175 cm, i.e., $H_0: \mu = 175 \, \text{cm}$.
   • Alternative Hypothesis (H₁): The mean height of adult males is not 175 cm, i.e., $H_1: \mu \neq 175 \, \text{cm}$ (this is a two-tailed test).

2. Significance Level:

   • The significance level is not directly given, but typically it's $\alpha = 0.05$, corresponding to a 95% confidence level. From the t-critical values given (-2.0452 to 2.0452), this aligns with a two-tailed test where $\alpha = 0.05$.

3. Degrees of Freedom (df):

   • The degrees of freedom for a one-sample t-test is calculated as: $df = n - 1$ where $n$ is the sample size. Since $n = 30$, the degrees of freedom are: $df = 30 - 1 = 29$

4. Decision Rule:

   • The decision rule involves comparing the calculated t-statistic to the critical values. The critical t-values at $\alpha = 0.05$ (two-tailed) are given as -2.0452 and 2.0452.
     • If $t$-statistic < -2.0452 or $t$-statistic > 2.0452, reject the null hypothesis.
     • Otherwise, do not reject the null hypothesis.

5. Calculate the Test Statistic:

   • The t-statistic is calculated using the formula: $t = \frac{\bar{x} - \mu}{\frac{s}{\sqrt{n}}}$ where:

     • $\bar{x} = 177 \, \text{cm}$ (sample mean),
     • $\mu = 175 \, \text{cm}$ (hypothesized population mean),
     • $s = 5 \, \text{cm}$ (sample standard deviation),
     • $n = 30$ (sample size).

     Substituting the values: $t = \frac{177 - 175}{\frac{5}{\sqrt{30}}} = \frac{2}{\frac{5}{5.477}} = \frac{2}{0.9129} = 2.19$

6. Result and Conclusion:

   • The calculated t-statistic is 2.19, which is greater than the critical value of 2.0452. Therefore, the test statistic falls in the rejection region.
   • Conclusion: Reject the null hypothesis. There is significant evidence to suggest that the mean height of the population is different from 175 cm at the 5% significance level.

Would you like further clarification or details?

Here are some questions to expand upon this topic:

1. What are the differences between one-sample t-tests and paired t-tests?
2. How does increasing the sample size affect the t-statistic in hypothesis testing?
3. What is the impact of using a one-tailed test versus a two-tailed test?
4. How would you calculate a confidence interval for the population mean in this scenario?
5. What assumptions must be met for a one-sample t-test to be valid?

Tip: Always ensure that the sample data is approximately normally distributed when using a one-sample t-test, especially for smaller sample sizes.