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 $\pm$ ). $\newline$ Answer:

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Algebra 2

Find probabilities using the normal distribution I

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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

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

\pm

\newline

Answer:

Identify given values: Identify the given values from the problem. $\newline$ Mean (sample mean) = $166$ pounds $\newline$ Standard deviation = $37$ pounds $\newline$ Sample size ( $n$ ) = $61$ $\newline$ Confidence level = $95\%$
Understand confidence level: Understand the $95\%$ confidence level in terms of the empirical rule. $\newline$ The empirical rule, also known as the $68-95-99.7$ rule, states that for a normal distribution: $\newline$ - Approximately $68\%$ of the data falls within $1$ standard deviation of the mean. $\newline$ - Approximately $95\%$ of the data falls within $2$ standard deviations of the mean. $\newline$ - Approximately $99.7\%$ of the data falls within $3$ standard deviations of the mean. $\newline$ Since we are interested in the $95\%$ confidence level, we will focus on the range within $2$ standard deviations of the mean.
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. $SEM = \frac{\text{Standard deviation}}{\sqrt{n}}$ $SEM = \frac{37}{\sqrt{61}}$
Perform SEM calculation: Perform the calculation for the standard error of the mean. $\newline$ $SEM = \frac{37}{\sqrt{61}}$ $\newline$ $SEM = \frac{37}{7.81025}$ (rounded to five decimal places) $\newline$ $SEM \approx 4.7369$ (rounded to four decimal places)
Determine margin of error: Determine the margin of error at the $95$ % confidence level using the empirical rule. $\newline$ Since the empirical rule states that $95$ % of the data falls within $2$ standard deviations of the mean, the margin of error is $2$ times the standard error of the mean. $\newline$ Margin of error = $2 \times \text{SEM}$ $\newline$ Margin of error = $2 \times 4.7369$
Calculate margin of error: Perform the calculation for the margin of error. $\newline$ Margin of error = $2 \times 4.7369$ $\newline$ Margin of error $\approx 9.4738$
Round margin of error: Round the margin of error to the nearest tenth. $\newline$ Margin of error $\approx 9.5$ (rounded to the nearest tenth)