Overweight Men For a random sample of 70 overweight men, the mean of the number of pounds that they were overwelght was 32 . The standard deviation of the population is 4.3 pounds. Part 1 of 4 (a) The best point estimate of the mean is 32 pounds. Part 2 of 4 (b) Find the $90 \%$ confidence interval of the mean of these pounds. Round intermediate answers to at least three decimal places. Round your final answers to one decimal place. \[ 31.2<\mu<32.8 \] Part 3 of 4 : (c) Find the $95 \%$ confidence interval of the mean of these pounds. Round intermediate answers to at least three decimal places. Round your final answers to one decimal place. \[ 31<\mu<33 \] Part: $3 / 4$ Part 4 of 4 (d) Which interval is larger? Why? The $\square \%$ confidence interval is larger. An interval with a Interval will be more likely to contain the true population

Step 1 ：The problem provides us with a sample of 70 overweight men, with a mean weight over the normal limit of 32 pounds and a standard deviation of 4.3 pounds.

Step 2 ：We are asked to find the 90% and 95% confidence intervals for the mean weight of these men. The formula for a confidence interval is \(\bar{x} \pm Z \frac{\sigma}{\sqrt{n}}\), where \(\bar{x}\) is the sample mean, \(Z\) is the Z-score (which depends on the confidence level), \(\sigma\) is the standard deviation of the population, and \(n\) is the sample size.

Step 3 ：For a 90% confidence level, the Z-score is approximately 1.645. Substituting the given values into the formula, we get the 90% confidence interval as approximately \((31.2, 32.8)\) pounds.

Step 4 ：For a 95% confidence level, the Z-score is approximately 1.96. Substituting the given values into the formula, we get the 95% confidence interval as approximately \((31, 33)\) pounds.

Step 5 ：The problem also asks us to compare the two intervals. The larger the confidence level, the larger the interval. This is because a larger confidence level means that we are more certain that the true population mean lies within the interval, so we need a larger interval to be more certain. Therefore, the 95% confidence interval is larger than the 90% confidence interval.

Step 6 ：Final Answer: The 90% confidence interval for the mean weight of the overweight men is approximately \(\boxed{(31.2, 32.8)}\) pounds and the 95% confidence interval is approximately \(\boxed{(31, 33)}\) pounds. The \(\boxed{95\%}\) confidence interval is larger.