Let's go through each part of the problem step by step.

Part 1: Degrees of Freedom for Finding the Critical Value $t_{\alpha/2}$

The degrees of freedom (df) for a t-distribution is calculated as: $\text{df} = n - 1$ where $n$ is the sample size.

Given:

• $n = 50$

So, the degrees of freedom are: $\text{df} = 50 - 1 = 49$

Answer: $\text{df} = 49$

Part 2: Critical Value $t_{\alpha/2}$ for a 95% Confidence Level

To find the critical value $t_{\alpha/2}$ for a 95% confidence level with 49 degrees of freedom, we can refer to the t-distribution table.

For a 95% confidence level:

• The value of $\alpha = 1 - 0.95 = 0.05$.
• The critical value $t_{\alpha/2}$ corresponds to $\alpha/2 = 0.025$.

Using the t-distribution table for $\text{df} = 49$ and $\alpha/2 = 0.025$, the critical value is approximately: $t_{\alpha/2} \approx 2.0096$

Answer: $t_{\alpha/2} \approx 2.01$ (rounded to two decimal places)

Part 3: General Description of the Number of Degrees of Freedom

Let's analyze the options given:

• Option A: "The number of degrees of freedom for a collection of sample data is the number of sample values that can vary after certain restrictions have been imposed on all data values."
  This is correct. Degrees of freedom refer to the number of values that can vary while calculating a statistic, such as the mean, after certain restrictions (like the sample mean) have been applied.

• Option B: "The number of degrees of freedom for a collection of sample data is the total number of sample values."
  This is incorrect because the degrees of freedom are generally the sample size minus the number of parameters estimated, not just the sample size.

• Option C: "The number of degrees of freedom for a collection of sample data is the number of unique, non-repeated sample values."
  This is incorrect because degrees of freedom are not dependent on the uniqueness or repetition of values.

• Option D: "The number of degrees of freedom for a collection of sample data is the number of sample values that are determined after certain restrictions have been imposed on all data values."
  This is incorrect because degrees of freedom refer to how many values are free to vary, not how many are determined.

Correct Answer:

• A. The number of degrees of freedom for a collection of sample data is the number of sample values that can vary after certain restrictions have been imposed on all data values.

Summary of Answers:

1. Degrees of Freedom: $\text{df} = 49$
2. Critical Value $t_{\alpha/2}$: $t_{\alpha/2} \approx 2.01$
3. General Description of Degrees of Freedom: A.

Would you like any further details or explanations on this problem?

Here are some related questions you might find helpful:

1. How does increasing the sample size affect the degrees of freedom?
2. What happens to the critical value if the confidence level is increased to 99%?
3. How is the t-distribution different from the normal distribution?
4. Why do we subtract one from the sample size to get the degrees of freedom?
5. How would the confidence interval change with a different confidence level?
6. What is the impact of a larger standard deviation on the confidence interval?
7. How do degrees of freedom affect the shape of the t-distribution?
8. What are the assumptions necessary to use the t-distribution for confidence intervals?

Tip: When using the t-distribution, always make sure to check the sample size and degrees of freedom, as they significantly affect the critical values and confidence intervals.