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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 36 residents and found the mean weight to be 182 pounds with a standard deviation of 29 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 3636 residents and found the mean weight to be 182182 pounds with a standard deviation of 2929 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 3636 residents and found the mean weight to be 182182 pounds with a standard deviation of 2929 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 given as 182182 pounds, the standard deviation (σ\sigma) is 2929 pounds, and the sample size (nn) is 3636 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=σnSEM = \frac{\sigma}{\sqrt{n}} SEM=2936SEM = \frac{29}{\sqrt{36}} SEM=296SEM = \frac{29}{6} SEM=4.8333...SEM = 4.8333... Rounded to the nearest tenth, SEM=4.8SEM = 4.8 pounds.
  3. Find Z-Score for 9595% Confidence: 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. Compute Margin of Error: Calculate the margin of error (ME). The margin of error is found by multiplying the z-score by the standard error of the mean. ME=z×SEMME = z \times SEM ME=1.96×4.8ME = 1.96 \times 4.8 ME=9.408ME = 9.408 Rounded to the nearest tenth, ME=9.4ME = 9.4 pounds.
  5. Determine Confidence Interval: Determine the confidence interval.\newlineThe confidence interval is calculated by adding and subtracting the margin of error from the sample mean.\newlineLower limit = xˉ−ME\bar{x} - ME\newlineUpper limit = xˉ+ME\bar{x} + ME\newlineLower limit = 182−9.4182 - 9.4\newlineUpper limit = 182+9.4182 + 9.4\newlineLower limit = 172.6172.6 pounds\newlineUpper limit = 191.4191.4 pounds

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