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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 61 residents and found the mean weight to be 166 pounds with a standard deviation of 37 pounds. At the 95% confidence level, use the normal distribution/empirical rule to estimate the margin of error for the mean, rounding to the nearest tenth. (Do not write +- ). Answer:

A study was commissioned to find the mean weight of the residents in certain town. The study examined a random sample of 6161 residents and found the mean weight to be 166166 pounds with a standard deviation of 3737 pounds. At the 95% 95 \% confidence level, use the normal distribution/empirical rule to estimate the margin of error for the mean, rounding to the nearest tenth. (Do not write ± \pm ).\newlineAnswer:

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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 6161 residents and found the mean weight to be 166166 pounds with a standard deviation of 3737 pounds. At the 95% 95 \% confidence level, use the normal distribution/empirical rule to estimate the margin of error for the mean, rounding to the nearest tenth. (Do not write ± \pm ).\newlineAnswer:
  1. Identify given values: Identify the given values from the problem.\newlineMean (sample mean) = 166166 pounds\newlineStandard deviation = 3737 pounds\newlineSample size (nn) = 6161\newlineConfidence level = 95%95\%
  2. Understand confidence level: Understand the 95%95\% confidence level in terms of the empirical rule.\newlineThe empirical rule, also known as the 68−95−99.768-95-99.7 rule, states that for a normal distribution:\newline- Approximately 68%68\% of the data falls within 11 standard deviation of the mean.\newline- Approximately 95%95\% of the data falls within 22 standard deviations of the mean.\newline- Approximately 99.7%99.7\% of the data falls within 33 standard deviations of the mean.\newlineSince we are interested in the 95%95\% confidence level, we will focus on the range within 22 standard deviations of the mean.
  3. Calculate standard error: Calculate the standard error of the mean (SEM). The standard error of the mean is the standard deviation divided by the square root of the sample size. SEM=Standard deviationnSEM = \frac{\text{Standard deviation}}{\sqrt{n}} SEM=3761SEM = \frac{37}{\sqrt{61}}
  4. Perform SEM calculation: Perform the calculation for the standard error of the mean. \newlineSEM=3761SEM = \frac{37}{\sqrt{61}}\newlineSEM=377.81025SEM = \frac{37}{7.81025} (rounded to five decimal places)\newlineSEM≈4.7369SEM \approx 4.7369 (rounded to four decimal places)
  5. Determine margin of error: Determine the margin of error at the 9595% confidence level using the empirical rule.\newlineSince the empirical rule states that 9595% of the data falls within 22 standard deviations of the mean, the margin of error is 22 times the standard error of the mean.\newlineMargin of error = 2×SEM2 \times \text{SEM}\newlineMargin of error = 2×4.73692 \times 4.7369
  6. Calculate margin of error: Perform the calculation for the margin of error.\newlineMargin of error = 2×4.73692 \times 4.7369\newlineMargin of error ≈9.4738\approx 9.4738
  7. Round margin of error: Round the margin of error to the nearest tenth.\newlineMargin of error ≈9.5\approx 9.5 (rounded to the nearest tenth)

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