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A study by the department of education of a certain state was trying to determine the mean SAT scores of the graduating high school seniors. The study examined the scores of a random sample of 129 graduating seniors and found the mean score to be 502 with a standard deviation of 92 . 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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A study by the department of education of a certain state was trying to determine the mean SAT scores of the graduating high school seniors. The study examined the scores of a random sample of $129$ graduating seniors and found the mean score to be $502$ with a standard deviation of $92$ . 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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Interpret confidence intervals for population means

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Q. A study by the department of education of a certain state was trying to determine the mean SAT scores of the graduating high school seniors. The study examined the scores of a random sample of

129

graduating seniors and found the mean score to be

502

with a standard deviation of

92

. Use the normal distribution/empirical rule to estimate a

95

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

Identify given information: Identify the given information. $\newline$ We have a sample mean ( $\bar{x}$ ) of $502$ , a standard deviation ( $\sigma$ ) of $92$ , and a sample size ( $n$ ) of $129$ . We want to estimate the $95\%$ confidence interval for the population mean.
Determine z-score: Determine the z-score that corresponds to a $95\%$ confidence level. $\newline$ For a $95\%$ confidence interval, the z-score that corresponds to the middle $95\%$ of the normal distribution is approximately $1.96$ . This value can be found in standard z-score tables or by using a calculator that provides the inverse cumulative distribution function for the standard normal distribution.
Calculate standard error: Calculate 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. $SEM = \frac{\sigma}{\sqrt{n}}$ $SEM = \frac{92}{\sqrt{129}}$ $SEM \approx \frac{92}{11.3578}$ $SEM \approx 8.1$ (rounded to the nearest tenth)
Calculate margin of error: Calculate the margin of error (ME). The margin of error is the product of the z-score and the standard error of the mean. $ME = z \times SEM$ $ME = 1.96 \times 8.1$ $ME \approx 15.876$ (rounded to the nearest tenth)
Calculate confidence interval bounds: Calculate the lower and upper bounds of the $95$ % confidence interval. $\newline$ Lower bound = $\bar{x} - ME$ $\newline$ Lower bound = $502 - 15.876$ $\newline$ Lower bound $\approx 486.1$ (rounded to the nearest tenth) $\newline$ Upper bound = $\bar{x} + ME$ $\newline$ Upper bound = $502 + 15.876$ $\newline$ Upper bound $\approx 517.9$ (rounded to the nearest tenth)
State final confidence interval: State the final $95\%$ confidence interval. $\newline$ The $95\%$ confidence interval for the mean SAT score is approximately $(486.1, 517.9)$ .

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