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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 32 students in the high school and found a mean savings of 4300 dollars with a standard deviation of 800 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 3232 students in the high school and found a mean savings of 43004300 dollars with a standard deviation of 800800 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 3232 students in the high school and found a mean savings of 43004300 dollars with a standard deviation of 800800 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. Identify Given Information: Identify the information given and what is being asked.\newlineWe are given:\newline- The mean savings of the sample (xˉ\bar{x}) = $4,300\$4,300\newline- The standard deviation of the sample (ss) = $800\$800\newline- The sample size (nn) = 3232 students\newline- The confidence level (95%95\%)\newlineWe need to find the margin of error for the mean savings at the 95%95\% confidence level.
  2. Concept of Margin of Error: Understand the concept of margin of error at the 95%95\% confidence level.\newlineThe margin of error at the 95%95\% confidence level can be found using the empirical rule, which states that approximately 95%95\% of the data falls within 22 standard deviations of the mean in a normal distribution.
  3. Calculate Standard Error: Calculate the standard error of the mean.\newlineThe standard error of the mean (SEM) is the standard deviation of the sampling distribution of the sample mean. It is calculated by dividing the standard deviation by the square root of the sample size.\newlineSEM = sn\frac{s}{\sqrt{n}}\newlineSEM = $80032\frac{\$800}{\sqrt{32}}\newlineSEM = $8005.656854249\frac{\$800}{5.656854249}\newlineSEM ≈\approx $141.421356237\$141.421356237
  4. Calculate Margin of Error: Calculate the margin of error using the empirical rule.\newlineSince we are using the empirical rule for a 9595% confidence level, we multiply the standard error by 22 (because 9595% of the data falls within 22 standard deviations of the mean).\newlineMargin of error = 2×SEM2 \times \text{SEM}\newlineMargin of error = 2×$(141.421356237)2 \times \$(141.421356237)\newlineMargin of error \approx \$(\(282\).\(842712474\))
  5. Round Margin of Error: Round the margin of error to the nearest whole number.\(\newline\)Margin of error \(\approx\) \(\$283\) (rounded to the nearest whole number)

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