Question

In Exercises 1–3, refer to the accompanying screen display that results from the Verizon airport data speeds (Mbps) from Data Set 32 “Airport Data Speeds” in Appendix B. The confidence level of 95% was used.

Degrees of Freedom

a. What is the number of degrees of freedom that should be used for finding the critical value $t_{\frac{\alpha }{2}}$?

b. Find the critical value $t_{\frac{\alpha }{2}}$corresponding to a 95% confidence level.

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

Short answer

a. The value of the degrees of freedom is equal to 49.

b. The critical value is equal to 2.0096.

c. The number of observations in the sample that can assume any value independently of other values stands for the number of degrees of freedom. Here, out of 50 values, 49 of them are independent values, while the last observation (50^th) is dependent on the values of the other 49 observations.

Step-by-step solution

Given Information

A sample of Verizon airport data speeds is considered.

Value of the degrees of freedom

The formula for computing the degrees of freedom is written below:

$df=n-1$

Here, n is the sample size and has a value equal to 50.

Substitute the value of n as follows:

$$\begin{gathered} df=50-1 \\ =49 \end{gathered}$$

Thus, the value of the degrees of freedom is equal to 49.

Critical value 

b.

The confidence level is equal to 95%. Thus, the corresponding level of significance is equal to 0.05.

Referring to the t-distribution table, the value of $t_{\frac{\alpha }{2}}$for 49 degrees of freedom when $\alpha$is equal to 0.05 is equal to 2.0096.

Meaning of the value of the degrees of freedom

c.

For a sample containing values, the number of independent values refers to the value of the degrees of freedom.

Here, out of 50 observations, 49 values can have any value that is assumed independently of other values. Based on the 49 observations, the 50^th observation can be calculated, and it is dependent on the value of the 49 observations.

Thus, the value of the degrees of freedom is equal to 49.