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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 212 graduating seniors and found the mean score to be 489 with a standard deviation of 107 . Use the normal distribution/empirical rule to estimate a 95% confidence interval for the mean, rounding all values to the nearest tenth. (◻,◻)

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 212 \mathbf{2 1 2} graduating seniors and found the mean score to be 489489 with a standard deviation of 107107 . Use the normal distribution/empirical rule to estimate a 9595\% confidence interval for the mean, rounding all values to the nearest tenth.

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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 212 \mathbf{2 1 2} graduating seniors and found the mean score to be 489489 with a standard deviation of 107107 . Use the normal distribution/empirical rule to estimate a 9595\% confidence interval for the mean, rounding all values to the nearest tenth.
  1. Identify given information: Identify the given information.\newlineWe have a sample mean (xˉ\bar{x}) of 489489, a standard deviation (σ\sigma) of 107107, and a sample size (nn) of 212212. We want to estimate the 95%95\% confidence interval for the population mean.
  2. Determine z-score: Determine the z-score that corresponds to a 95%95\% confidence level.\newlineFor a 95%95\% confidence interval, the z-score that corresponds to the middle 95%95\% of the normal distribution is approximately 1.961.96. This value can be found in standard z-score tables or by using a calculator that provides the z-score for a given confidence level.
  3. Calculate SEM: 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=σnSEM = \frac{\sigma}{\sqrt{n}} SEM=107212SEM = \frac{107}{\sqrt{212}}
  4. Calculate standard error: Perform the calculation for the standard error of the mean. \newlineSEM=107212SEM = \frac{107}{\sqrt{212}}\newlineSEM≈10714.56SEM \approx \frac{107}{14.56}\newlineSEM≈7.35SEM \approx 7.35 (rounded to the nearest tenth)
  5. Calculate margin of error: Calculate the margin of error (ME) for the confidence interval.\newlineThe margin of error is found by multiplying the z-score by the standard error of the mean.\newlineME=z×SEMME = z \times SEM\newlineME=1.96×7.35ME = 1.96 \times 7.35
  6. Calculate lower and upper bounds: Perform the calculation for the margin of error.\newlineME=1.96×7.35ME = 1.96 \times 7.35\newlineME≈14.41ME \approx 14.41 (rounded to the nearest tenth)
  7. Calculate lower and upper bounds: Perform the calculation for the margin of error.\newlineME=1.96×7.35ME = 1.96 \times 7.35\newlineME≈14.41ME \approx 14.41 (rounded to the nearest tenth)Calculate the lower and upper bounds of the 9595% confidence interval.\newlineLower bound = xˉ−ME\bar{x} - ME\newlineUpper bound = xˉ+ME\bar{x} + ME\newlineLower bound = 489−14.41489 - 14.41\newlineUpper bound = 489+14.41489 + 14.41
  8. Calculate lower and upper bounds: Perform the calculation for the margin of error.\newlineME=1.96×7.35ME = 1.96 \times 7.35\newlineME≈14.41ME \approx 14.41 (rounded to the nearest tenth)Calculate the lower and upper bounds of the 9595% confidence interval.\newlineLower bound = xˉ−ME\bar{x} - ME\newlineUpper bound = xˉ+ME\bar{x} + ME\newlineLower bound = 489−14.41489 - 14.41\newlineUpper bound = 489+14.41489 + 14.41Perform the calculations for the lower and upper bounds.\newlineLower bound ≈489−14.41\approx 489 - 14.41\newlineLower bound ≈474.6\approx 474.6 (rounded to the nearest tenth)\newlineUpper bound ≈489+14.41\approx 489 + 14.41\newlineUpper bound ≈503.4\approx 503.4 (rounded to the nearest tenth)

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