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A study was commissioned to find the mean weight of the residents in certain town. The study examined a random sample of 133 residents and found the mean weight to be 165 pounds with a standard deviation of 25 pounds. Use the normal distribution/empirical rule to estimate a 95% confidence interval for the mean, rounding all values to the nearest tenth.

A study was commissioned to find the mean weight of the residents in certain town. The study examined a random sample of 133133 residents and found the mean weight to be 165165 pounds with a standard deviation of 2525 pounds. Use the normal distribution/empirical rule to estimate a 95% 95 \% confidence interval for the mean, rounding all values to the nearest tenth.

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Q. A study was commissioned to find the mean weight of the residents in certain town. The study examined a random sample of 133133 residents and found the mean weight to be 165165 pounds with a standard deviation of 2525 pounds. Use the normal distribution/empirical rule to estimate a 95% 95 \% confidence interval for the mean, rounding all values to the nearest tenth.
  1. Identify Data Parameters: Identify the sample mean, standard deviation, and sample size. The sample mean (xˉ\bar{x}) is 165165 pounds, the standard deviation (ss) is 2525 pounds, and the sample size (nn) is 133133 residents.
  2. Calculate Standard Error: Determine the standard error of the mean (SEM). The standard error of the mean is calculated by dividing the standard deviation by the square root of the sample size. SEM=snSEM = \frac{s}{\sqrt{n}} SEM=25133SEM = \frac{25}{\sqrt{133}} SEM≈2511.5326SEM \approx \frac{25}{11.5326} SEM≈2.1689SEM \approx 2.1689 Round to the nearest tenth: SEM≈2.2SEM \approx 2.2 pounds
  3. Find Z-Score: Find the z-score that corresponds to a 95%95\% confidence level.\newlineFor a 95%95\% confidence interval, the z-score is typically 1.961.96 (this value comes from standard normal distribution tables).
  4. Calculate Margin of Error: Calculate the margin of error (ME) using the z-score and the standard error.\newlineME=z×SEMME = z \times SEM\newlineME=1.96×2.2ME = 1.96 \times 2.2\newlineME≈4.312ME \approx 4.312\newlineRound to the nearest tenth: ME≈4.3ME \approx 4.3 pounds
  5. Determine Confidence Interval Bounds: Calculate the lower and upper bounds of the 9595% confidence interval.\newlineLower bound = xˉ−ME\bar{x} - ME\newlineLower bound = 165−4.3165 - 4.3\newlineLower bound ≈160.7\approx 160.7 pounds\newlineUpper bound = xˉ+ME\bar{x} + ME\newlineUpper bound = 165+4.3165 + 4.3\newlineUpper bound ≈169.3\approx 169.3 pounds
  6. Final Confidence Interval: Round the lower and upper bounds to the nearest tenth and state the final 95%95\% confidence interval.\newlineThe 95%95\% confidence interval for the mean weight of the residents is approximately (160.7,169.3)(160.7, 169.3) pounds.

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