Questions: Overweight Men For a random sample of 65 overweight men, the mean of the number of pounds that they were overweight was 29. The standard deviation of the population is 4.5 pounds. Part 1 of 4 (a) The best point estimate of the mean is 29 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. 28.1<μ<29.9 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. 27.9<μ<30.1 Part 4 of 4 (d) Which interval is larger? Why? The 99% confidence interval is larger. An interval with a greater range of values than the 99% confidence interval will be more likely to contain the true population mean.

Overweight Men For a random sample of 65 overweight men, the mean of the number of pounds that they were overweight was 29. The standard deviation of the population is 4.5 pounds. Part 1 of 4 (a) The best point estimate of the mean is 29 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. 28.1<μ<29.9 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. 27.9<μ<30.1 Part 4 of 4 (d) Which interval is larger? Why? The 99% confidence interval is larger. An interval with a greater range of values than the 99% confidence interval will be more likely to contain the true population mean.
Transcript text: Overweight Men For a random sample of 65 overweight men, the mean of the number of pounds that they were overweight was 29 . The standard deviation of the population is 4.5 pounds. Part 1 of 4 (a) The best point estimate of the mean is 29 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. \[ 28.1<\mu<29.9 \] 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. \[ 27.9<\mu<30.1 \] Part: 3 / 4 Part 4 of 4 (d) Which interval is larger? Why? The $99 \%$ confidence interval is larger. An interval with a greater range of values than the $99 \%$ confidence interval will be more likely to contain the true population mean. Try again Skip Part Recheck Submit Assignment
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Solution

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Solution Steps

Step 1: Point Estimate

The best point estimate of the mean number of pounds that the overweight men were overweight is given by the sample mean:

\[ \bar{x} = 29 \text{ pounds} \]

Step 2: 90% Confidence Interval

To calculate the 90% confidence interval for the mean, we use the formula:

\[ \bar{x} \pm z \frac{\sigma}{\sqrt{n}} \]

Where:

  • \(\bar{x} = 29\)
  • \(z\) for 90% confidence level is approximately \(1.645\)
  • \(\sigma = 4.5\)
  • \(n = 65\)

Calculating the margin of error:

\[ \text{Margin of Error} = 1.645 \cdot \frac{4.5}{\sqrt{65}} \approx 1.6 \]

Thus, the 90% confidence interval is:

\[ (29 - 1.6, 29 + 1.6) = (28.1, 29.9) \]

Step 3: 95% Confidence Interval

For the 95% confidence interval, we again use the formula:

\[ \bar{x} \pm z \frac{\sigma}{\sqrt{n}} \]

Where:

  • \(z\) for 95% confidence level is approximately \(1.96\)

Calculating the margin of error:

\[ \text{Margin of Error} = 1.96 \cdot \frac{4.5}{\sqrt{65}} \approx 2.0 \]

Thus, the 95% confidence interval is:

\[ (29 - 2.0, 29 + 2.0) = (27.9, 30.1) \]

Final Answer

The confidence intervals are:

  • 90% Confidence Interval: \(\boxed{(28.1, 29.9)}\)
  • 95% Confidence Interval: \(\boxed{(27.9, 30.1)}\)
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