A study was commissioned to find the mean weight of the residents in certain town. The study examined a random sample of $113$ residents and found the mean weight to be $159$ pounds with a standard deviation of $34$ 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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Q. A study was commissioned to find the mean weight of the residents in certain town. The study examined a random sample of

113

residents and found the mean weight to be

159

pounds with a standard deviation of

34

pounds. Use the normal distribution/empirical rule to estimate a

95 \%

confidence interval for the mean, rounding all values to the nearest tenth.

Find Z-score: To estimate a $95$ % confidence interval for the mean weight using the normal distribution, we will use the formula for the confidence interval: $\newline$ Mean $\pm$ $Z$ -score $($ Standard Deviation / \sqrt{Sample Size}\)) $\newline$ First, we need to find the $Z$ -score that corresponds to a $95$ % confidence level.
Calculate Standard Error: For a $95\%$ confidence interval, the Z-score is typically $1.96$ . This value comes from the standard normal distribution table, which provides the Z-score that corresponds to the desired confidence level.
Calculate Margin of Error: Next, we calculate the standard error of the mean by dividing the standard deviation by the square root of the sample size. $\newline$ Standard Error = Standard Deviation / $\sqrt{\text{Sample Size}}$ $\newline$ Standard Error = $\frac{34}{\sqrt{113}}$
Calculate Lower and Upper Bounds: Now we perform the calculation for the standard error. $\newline$ Standard Error = $\frac{34}{\sqrt{113}} \approx \frac{34}{10.6301} \approx 3.2$
Calculate Lower and Upper Bounds: Now we perform the calculation for the standard error. $\newline$ Standard Error = $\frac{34}{\sqrt{113}} \approx \frac{34}{10.6301} \approx 3.2$ With the standard error calculated, we can now find the margin of error by multiplying the Z-score by the standard error. $\newline$ Margin of Error = Z-score $\times$ Standard Error $\newline$ Margin of Error = $1.96 \times 3.2$
Calculate Lower and Upper Bounds: Now we perform the calculation for the standard error. $\newline$ Standard Error = $\frac{34}{\sqrt{113}} \approx \frac{34}{10.6301} \approx 3.2$ With the standard error calculated, we can now find the margin of error by multiplying the Z-score by the standard error. $\newline$ Margin of Error = Z-score $\times$ Standard Error $\newline$ Margin of Error = $1.96 \times 3.2$ We calculate the margin of error. $\newline$ Margin of Error $\approx 1.96 \times 3.2 \approx 6.272$
Calculate Lower and Upper Bounds: Now we perform the calculation for the standard error. $\newline$ Standard Error = $\frac{34}{\sqrt{113}} \approx \frac{34}{10.6301} \approx 3.2$ With the standard error calculated, we can now find the margin of error by multiplying the Z-score by the standard error. $\newline$ Margin of Error = Z-score $\times$ Standard Error $\newline$ Margin of Error = $1.96 \times 3.2$ We calculate the margin of error. $\newline$ Margin of Error $\approx 1.96 \times 3.2 \approx 6.272$ Finally, we apply the margin of error to the mean weight to find the confidence interval. $\newline$ Lower Bound = Mean - Margin of Error $\newline$ Upper Bound = Mean + Margin of Error $\newline$ Lower Bound $\approx 159 - 6.272$ $\newline$ Upper Bound $\approx 159 + 6.272$
Calculate Lower and Upper Bounds: Now we perform the calculation for the standard error. $\newline$ Standard Error = $\frac{34}{\sqrt{113}} \approx \frac{34}{10.6301} \approx 3.2$ With the standard error calculated, we can now find the margin of error by multiplying the Z-score by the standard error. $\newline$ Margin of Error = Z-score $\times$ Standard Error $\newline$ Margin of Error = $1.96 \times 3.2$ We calculate the margin of error. $\newline$ Margin of Error $\approx 1.96 \times 3.2 \approx 6.272$ Finally, we apply the margin of error to the mean weight to find the confidence interval. $\newline$ Lower Bound = Mean - Margin of Error $\newline$ Upper Bound = Mean + Margin of Error $\newline$ Lower Bound $\approx 159 - 6.272$ $\newline$ Upper Bound $\approx 159 + 6.272$ We calculate the lower and upper bounds of the confidence interval, rounding to the nearest tenth. $\newline$ Lower Bound $\approx 159 - 6.272 \approx 152.7$ $\newline$ Upper Bound $\approx 159 + 6.272 \approx 165.3$