Refer to the accompanying data display that results from a sample of airport data speeds in Mbps. Complete parts​ (a) through​ (c) below.

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TInterval

​(13.046,22.15)

x overbarxequals=17.598

Sxequals=16.01712719

nequals=50

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Part 1

**a.** What is the number of degrees of freedom that should be used for finding the critical value 

t Subscript alpha divided by 2tα/2​?

dfequals=enter your response here

​(Type a whole​ number.)

Part 2

**b.** Find the critical value 

t Subscript alpha divided by 2tα/2 

corresponding to a​ 95% confidence level.

t Subscript alpha divided by 2tα/2equals=enter your response here

​(Round to two decimal places as​ needed.)

Part 3

**c.** Give a brief general description of the number of degrees of freedom.

A.

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.

B.

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.

C.

The number of degrees of freedom for a collection of sample data is the total number of sample values.

D.

The number of degrees of freedom for a collection of sample data is the number of​ unique, non-repeated sample values.

### Part 1: Determine Degrees of Freedom

The degrees of freedom (df) for a sample is given by:
\[ df = n - 1 \]

Given that the sample size \( n = 50 \):
\[ df = 50 - 1 = 49 \]

So, the number of degrees of freedom is:
\[ \boxed{49} \]

### Part 2: Find the Critical Value \( t_{\alpha/2} \)

To find the critical value \( t_{\alpha/2} \) for a 95% confidence level with 49 degrees of freedom, we refer to the t distribution table or use a calculator for the t distribution.

For a 95% confidence level, \( \alpha = 1 - 0.95 = 0.05 \). Thus, \( \alpha/2 = 0.025 \).

Using the t distribution table for \( df = 49 \) and \( \alpha/2 = 0.025 \), the critical value \( t_{\alpha/2} \) is:
\[ \boxed{2.01} \]

### Part 3: Describe Degrees of Freedom

The correct description of the number of degrees of freedom is:

B. 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
1. **Degrees of freedom (df):** \( \boxed{49} \)
2. **Critical value \( t_{\alpha/2} \):** \( \boxed{2.01} \)
3. **Description of degrees of freedom:** B. 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.

Do you want any details or have any questions?

Here are some related questions to expand your understanding:

1. How does the number of degrees of freedom affect the shape of the t distribution?
2. What happens to the critical value \( t_{\alpha/2} \) as the degrees of freedom increase?
3. Why do we use the t distribution instead of the normal distribution for small sample sizes?
4. How can you use the t distribution table to find critical values for other confidence levels?
5. What are some common applications of the t distribution in statistics?
6. How would the confidence interval change if the sample size was larger?
7. Why is it important to understand degrees of freedom in hypothesis testing?
8. What are some limitations of using the t distribution for confidence intervals?

**Tip:** When using statistical tables, always ensure you are referencing the correct table for your specific degrees of freedom and confidence level.

Explore the concept of degrees of freedom and t-distribution critical values in statistics. Learn how to determine degrees of freedom, find critical values for confidence intervals, and understand their significance in statistical analysis. Suitable for college-level students and professionals.

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