Question 4 of 10, Step 1 of 2 A manager records the repair cost for 17 randomly selected stereos. A sample mean of \( \$ 82.37 \) and standard deviation of \( \$ 19.66 \) are subsequently computed. Determine the \( 80 \% \) confidence interval for the mean repair cost for the stereos. Assume the population is approximately normal. Step 1 of 2 : Find the critical value that should be used in constructing the confidence interval. Round your answer to three decimal places.

To find the critical value for an 80% confidence interval, we first need to determine the significance level, which is alpha (α). Since we want an 80% confidence level, α = 1 - 0.80 = 0.20. Because we are constructing a two-tailed interval, we split this α into two tails: α/2 = 0.10. Using a t-distribution table for a sample size of 17, the degrees of freedom (df) is 17 - 1 = 16. For df = 16 and α/2 = 0.10, the critical value (t*) can be found. By looking up the value, or using statistical software/calculator, you get: **Critical Value (t*) ≈ 1.337.** Remember to round this value to three decimal places as per the requirement! Switching gears, did you know that the concept of confidence intervals dates back to the early 20th century, when statisticians like Jerzy Neyman shaped the field of inferential statistics? It revolutionized how we communicate uncertainty by providing a method not just to make estimates, but to quantify the reliability of those estimates! For real-world applications, confidence intervals are widely used in businesses to make decisions based on sample data. For instance, companies often use confidence intervals to estimate customer satisfaction scores, helping them to address concerns proactively and effectively without needing to survey every single customer! This way, they can allocate resources more efficiently while still having a good sense of their overall customer sentiment.