Question 3 of 10, Step 1 of 2 A consumer affairs investigator records the repair cost for 22 randomly selected TVs: A sample mean of \( \$ 83.23 \) and standard deviation of \( \$ 22.67 \) are subsequently computed. Determine the \( 95 \% \) confidence interval for the mean repair cost for the TVs. 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 determine the critical value for a 95% confidence interval, we need to look up the z-score that corresponds to the tail probabilities of 0.025 in each tail (since 1 - 0.95 = 0.05, and we split this between two tails). For a standard normal distribution, the critical z-value for 95% confidence is approximately \( 1.96 \). Now that we've established the critical z-value, it will aid in calculating the margin of error. This means whenever you need to gauge the precision of estimates, having a solid understanding of critical values will empower your analysis and lead to more confident conclusions about your data. Did you know that the concept of confidence intervals originated with statisticians like Jerzy Neyman in the 1930s? Neyman's work laid the foundation for modern statistical inference, which revolutionized data analysis in many fields, including psychology, medicine, and marketing. It's amazing to think that just by analyzing some random samples, we can make educated guesses about broader populations!