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A study of a local high school tried to determine the mean amount of money that each student had saved. The study surveyed a random sample of 59 students in the high school and found a mean savings of 4000 dollars with a standard deviation of 1000 dollars. At the 95% confidence level, use the normal distribution/empirical rule to estimate the margin of error for the mean, rounding to the nearest whole number. (Do not write +- ). Answer:

A study of a local high school tried to determine the mean amount of money that each student had saved. The study surveyed a random sample of 5959 students in the high school and found a mean savings of 40004000 dollars with a standard deviation of 10001000 dollars. At the 95% 95 \% confidence level, use the normal distribution/empirical rule to estimate the margin of error for the mean, rounding to the nearest whole number. (Do not write ± \pm ).\newlineAnswer:

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Q. A study of a local high school tried to determine the mean amount of money that each student had saved. The study surveyed a random sample of 5959 students in the high school and found a mean savings of 40004000 dollars with a standard deviation of 10001000 dollars. At the 95% 95 \% confidence level, use the normal distribution/empirical rule to estimate the margin of error for the mean, rounding to the nearest whole number. (Do not write ± \pm ).\newlineAnswer:
  1. Understand the problem: Understand the problem and the data given.\newlineWe are given the mean savings of a sample of students ($4000\$4000), the standard deviation of the savings ($1000\$1000), and the sample size (5959 students). We are asked to find the margin of error at the 95%95\% confidence level using the normal distribution.
  2. Identify z-score: Identify the z-score associated with the 95%95\% confidence level.\newlineFor a 95%95\% confidence level, 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 z-score tables or using statistical software.
  3. Calculate SEM: Calculate the standard error of the mean.\newlineThe standard error of the mean (SEM) is the standard deviation divided by the square root of the sample size. The formula is SEM=SDn\text{SEM} = \frac{\text{SD}}{\sqrt{n}}.\newlineSEM=100059\text{SEM} = \frac{1000}{\sqrt{59}}
  4. Calculate SEM: Perform the calculation for the standard error of the mean. \newlineSEM=100059SEM = \frac{1000}{\sqrt{59}}\newlineSEM≈10007.68SEM \approx \frac{1000}{7.68}\newlineSEM≈130.21SEM \approx 130.21
  5. Calculate ME: Calculate the margin of error using the z-score and the standard error.\newlineThe margin of error (ME) is the z-score multiplied by the standard error of the mean. The formula is ME=z×SEMME = z \times SEM.\newlineME=1.96×130.21ME = 1.96 \times 130.21
  6. Calculate ME: Perform the calculation for the margin of error.\newlineME=1.96×130.21ME = 1.96 \times 130.21\newlineME≈255.21ME \approx 255.21
  7. Round ME: Round the margin of error to the nearest whole number.\newlineRounded ME ≈255\approx 255

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