Question 7 of 10, step 1 of 2 In a random sample of 11 residents of the state of Florida, the mean waste recycled per person per day was 2.9 pounds with a standard deviation of 0.42 pounds. Determine the \( 95 \% \) confidence interval for the mean waste recycled per person per day for the population of Florida. 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 when dealing with a small sample size (n = 11) and an approximately normal distribution, you'll want to use the t-distribution. For a \(95\%\) confidence level and \(n - 1\) degrees of freedom (which is \(10\) in this case), you can look up the t-value in a t-distribution table or use statistical software. The critical t-value for \(10\) degrees of freedom at a \(95\%\) confidence level is approximately \(2.228\) when rounded to three decimal places. Now, let’s groove into a little background on those confidence intervals! This statistical tool helps us estimate a population parameter by providing a range that likely includes it. You may have noticed that as the sample size increases, the confidence intervals tend to narrow down—providing a more precise estimate. It’s like using a telescope to get a clear view of the stars; the more you look, the more details you can see! To spice things up even more, consider the real-world applications of confidence intervals. They’re not just numbers on a piece of paper! Businesses often use them to assess consumer behavior or product performance. For instance, a company might want to determine how much garbage its consumers are recycling. By conducting a small survey and analyzing the data, they can confidently estimate the average recycling behavior of their entire customer base—and make those eco-friendly decisions that matter!