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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 $36$ 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.

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Use normal distributions to approximate binomial distributions

Full solution

Q. 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.

Identify Data Parameters: Identify the sample mean, standard deviation, and sample size. The sample mean ( $\bar{x}$ ) is given as $182$ pounds, the standard deviation ( $\sigma$ ) is $29$ pounds, and the sample size ( $n$ ) is $36$ residents.
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. $SEM = \frac{\sigma}{\sqrt{n}}$ $SEM = \frac{29}{\sqrt{36}}$ $SEM = \frac{29}{6}$ $SEM = 4.8333...$ Rounded to the nearest tenth, $SEM = 4.8$ pounds.
Find Z-Score for $95$ % Confidence: Find the z-score that corresponds to a $95\%$ confidence level. $\newline$ For a $95\%$ confidence interval, the z-score is typically $1.96$ (this value comes from standard normal distribution tables).
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. $ME = z \times SEM$ $ME = 1.96 \times 4.8$ $ME = 9.408$ Rounded to the nearest tenth, $ME = 9.4$ pounds.
Determine Confidence Interval: Determine the confidence interval. $\newline$ The confidence interval is calculated by adding and subtracting the margin of error from the sample mean. $\newline$ Lower limit = $\bar{x} - ME$ $\newline$ Upper limit = $\bar{x} + ME$ $\newline$ Lower limit = $182 - 9.4$ $\newline$ Upper limit = $182 + 9.4$ $\newline$ Lower limit = $172.6$ pounds $\newline$ Upper limit = $191.4$ pounds

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