Overweight Men For a random sample of 60 overweight men, the mean of the number of pounds that they were overweight was 29 . The standard deviation o the population is 3.9 pounds. Part 1 of 4 (a) The best point estimate of the mean is 29 pounds. Part: 1 / 4 Part 2 of 4 (b) 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. ( 28.0 < \( \mu<30.1 \)

To calculate the 95% confidence interval for the mean, we can use the formula: \[ \text{CI} = \bar{x} \pm z \left(\frac{\sigma}{\sqrt{n}}\right) \] Where: - \(\bar{x} = 29\) (sample mean) - \(\sigma = 3.9\) (population standard deviation) - \(n = 60\) (sample size) - \(z\) is the z-value for a 95% confidence level, which is approximately 1.96. Calculating the standard error (SE): \[ SE = \frac{\sigma}{\sqrt{n}} = \frac{3.9}{\sqrt{60}} \approx 0.503 \] Now, calculating the margin of error (ME): \[ ME = z \times SE \approx 1.96 \times 0.503 \approx 0.986 \] Finally, the confidence interval will be: \[ \text{CI} = 29 \pm 0.986 \] Which results in: \[ (28.0, 30.0) \] So, the 95% confidence interval for the mean number of pounds overweight is approximately \( (28.0, 30.0) \). --- Historical Background: The concept of confidence intervals dates back to the 1930s, a brainchild of statisticians like Jerzy Neyman. Neyman introduced the idea to give researchers a way to express the uncertainty of their estimates instead of just providing a single number. This fundamental shift allowed for a more robust understanding of data, essentially shaping modern statistical analysis. Real-World Application: In healthcare and fitness research, confidence intervals help practitioners evaluate the effectiveness of diets or exercise programs. For instance, knowing the average weight loss with a diet plan might come with a confidence interval, allowing nutritionists to gauge how reliable the results are. This insight can guide individuals in making informed decisions about their health journey.