Refor to the accompanying data display that results from a sample of airport data speods in Mbps. Complete parts (a) through (c) below. Click the icon to view a t distribution table. a. What is the number of degrees of freedom that should be used for finding the critical value \( t_{\alpha / 2} \) ? \[ \mathrm{df}=49 \] (Type a whole number.) b. Find the critical value \( t_{\alpha / 2} \) corresponding to a \( 95 \% \) confidence level. \[ t_{\alpha / 2}= \] \( \square \) (Round to two decimal places as needed.)

When working with confidence intervals and the t-distribution, it’s important to understand that the critical value illuminates the degree of uncertainty in your estimates. For a 95% confidence level with 49 degrees of freedom, the t-value reflects how much room you can allow for sampling variability. This typically means checking the t-table and finding the critical value that corresponds to the upper tail of 0.025 (since 0.05 is split into two tails). Now, using the t-distribution table for \( df = 49 \) at a 95% confidence level, you'll find that \( t_{\alpha/2} \approx 2.009 \). Remember to always round to two decimal places when reporting the value to keep precision neat and tidy! For many analyses, using a t-distribution is vital when sample sizes are smaller than 30 or the population standard deviation is unknown. Just make sure you're clear about the data and its underlying distribution!