To determine the confidence interval for the population mean (μ) based on your test results, we need to use the t-distribution because you're conducting a single sample t-test.

Given your information:

• Test statistic (t) = 1.968376
• Degrees of freedom (df) = 21
• p-value = 0.06236669
• Significance level (α) = 0.05

Confidence Interval Calculation

1. Find the critical t-value: For a 95% confidence interval and 21 degrees of freedom, look up the critical t-value. This is approximately ±2.080 (you can use a t-table or statistical software).

2. Calculate the standard error (SE): If you have the sample standard deviation (s) and sample size (n), you can calculate SE using: $SE = \frac{s}{\sqrt{n}}$

3. Calculate the margin of error (ME): $ME = t_{\alpha/2} \times SE$

4. Construct the confidence interval: $\text{Confidence Interval} = \bar{x} \pm ME$ Where $\bar{x}$ is the sample mean.

Interpretation

With a confidence interval, we are stating that we are 95% confident that the true population mean (μ) lies within the calculated interval. If the interval does not include $50, it suggests that the population mean could be different from this value.

Your Statement

Since you mentioned the interval as "22 < μ < 50" but labeled it incorrect, you will need to correctly calculate the confidence interval based on the sample data.

Conclusion

1. Is it reasonable to conclude that the population mean is $50? Yes, since we fail to reject H0, it's reasonable, but it's still uncertain based on the confidence interval.

2. Type of error: Since you failed to reject H0, a Type I error (rejecting a true null hypothesis) is not applicable. However, a Type II error (failing to reject a false null hypothesis) could be possible, given that the p-value is close to the significance level.

Questions for Further Exploration

1. What is the sample size (n) and sample standard deviation (s) used in your test?
2. Can you provide the sample mean ($\bar{x}$) to calculate the confidence interval accurately?
3. How would you interpret a confidence interval that includes $50 versus one that does not?
4. What are the implications of failing to reject H0 in your analysis?
5. How would the results change if you had set a different significance level (e.g., α = 0.01)?

Tip

Always verify your calculations for the confidence interval, as it can provide critical insights into the validity of your hypothesis testing results.