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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Identify given information: Identify the given information. We have a sample mean (x̄) of 502, a standard deviation (σ) 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. 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 = σ / √n SEM = 92 / √129 SEM ≈ 92 / 11.3578 SEM ≈ 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 * SEM ME = 1.96 * 8.1 ME ≈ 15.876 (rounded to the nearest tenth)Calculate confidence interval bounds: Calculate the lower and upper bounds of the 95% confidence interval. Lower bound = x̄ - ME Lower bound = 502 - 15.876 Lower bound ≈ 486.1 (rounded to the nearest tenth)  Upper bound = x̄ + ME Upper bound = 502 + 15.876 Upper bound ≈ 517.9 (rounded to the nearest tenth)State final confidence interval: State the final 95% confidence interval. The 95% confidence interval for the mean SAT score is approximately (486.1, 517.9).

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