Constructing Confidence Intervals In Exercises 21 and 22, you are given the sample mean and the sample standard deviation. Assume the random variable is normally distributed and use a normal distribution or a t-distribution to construct a 90\% confidence interval for the population mean. If convenient, use technology to construct the confidence interval. 21. Waste Generated States, the mean waste generated per person per day was 4.50 pounds and the standard deviation was 1.21 pounds. (b) Repeat part (a), assuming the same statistics came from a sample size of 500. Compare the results. (Adapted from

To construct a 90% confidence interval for the population mean when the sample size is 500, we can use the t-distribution or normal distribution. With a sample mean (\( \bar{x} \)) of 4.50 pounds and a standard deviation (s) of 1.21 pounds, you can find the margin of error using the formula: \[ \text{Margin of Error} = z^* \frac{s}{\sqrt{n}} \] For a 90% confidence level and a large sample size (n=500), we use the z-distribution (z* ≈ 1.645). Thus, \[ \text{Margin of Error} = 1.645 \cdot \frac{1.21}{\sqrt{500}} \approx 0.092 \] Now, the confidence interval would be: \[ (4.50 - 0.092, 4.50 + 0.092) \Rightarrow (4.408, 4.592) \] When the sample size is larger, the confidence interval becomes narrower because the standard error decreases, indicating more precision in estimations. For further understanding, you might find it interesting that increasing the sample size reduces the margin of error, making your estimation of the population mean more accurate! Consider practical scenarios where having more data can lead to better decision-making, like in healthcare research or market analysis, where precision is key!