# Statistics - Standard Deviation

The average math SAT score is 500 with a standard deviation of 100. A particular high school claims that its students have unusually high math SAT scores. A random sample of 50 students from this school was selected, and the mean math SAT score was 530. Is the high school justified in its claim

### Comments for Statistics - Standard Deviation

 Jul 10, 2011 Statistics – Margin of Error by: Staff -------------------------------------------------------------- Part II At the 90% confidence level: Z (90% confidence level) = 1.645 σ (standard deviation) = 100 (given in statement of problem) N (sample size) = 50 (given in statement of problem) Margin of Error at 90% confidence level = ± (1.645)*( 100)/[sqrt(50)] Margin of Error at 90% confidence level = ± (1.645)*( 100)/(7.071) Margin of Error at 90% confidence level = ± 23.264 Actual SAT Average = calculated average ± margin of error Actual SAT Average (90% conf) = 530 ± 23.264 Actual SAT Average (90% conf) = somewhere between 507 to 553 At the 95% confidence level: Z (95% confidence level) = 1.96 σ (standard deviation) = 100 (given in statement of problem) N (sample size) = 50 (given in statement of problem) Margin of Error at 95% confidence level = ± (1.96)*( 100)/[sqrt(50)] Margin of Error at 95% confidence level = ± (1.96)*( 100)/(7.071) Margin of Error at 95% confidence level = ± 27.719 Actual SAT Average = calculated average ± margin of error Actual SAT Average (95% conf) = 530 ± 27.719 Actual SAT Average (95% conf) = somewhere between 502 to 558 At the 99% confidence level: Z (99% confidence level) = 2.58 σ (standard deviation) = 100 (given in statement of problem) N (sample size) = 50 (given in statement of problem) Margin of Error at 99% confidence level = ± (2.58)*( 100)/[sqrt(50)] Margin of Error at 99% confidence level = ± (2.58)*( 100)/(7.071) Margin of Error at 99% confidence level = ± 36.487 Actual SAT Average = calculated average ± margin of error Actual SAT Average (99% conf) = 530 ± 36.487 Actual SAT Average (99% conf) = somewhere between 494 to 566 Summary of results: Actual SAT Average (90% conf) = somewhere between 507 to 553 Actual SAT Average (95% conf) = somewhere between 502 to 558 Actual SAT Average (99% conf) = somewhere between 494 to 566 The final answer: Is the high school justified in its claim? NO At the 99% confidence level, the actual school average score could be anywhere between 494 and 566. Note: the margin of error can only be decreased by taking a larger sample. Thanks for writing. Staff www.solving-math-problems.com

 Jul 10, 2011 Statistics – Margin of Error by: Staff Part I The question: The average math SAT score is 500 with a standard deviation of 100. A particular high school claims that its students have unusually high math SAT scores. A random sample of 50 students from this school was selected, and the mean math SAT score was 530. Is the high school justified in its claim The answer: To justify the school’s claim, the average SAT score of 530 must be a true and reliable statistic. If another “random sample” is taken and averaged, would the new SAT average score also be 530? Or . . . Is there a margin of error which must be taken into account? The answer is YES. There is always a margin or error which must be taken into account. This can be expressed mathematically as follows: Actual SAT Average = calculated average ± margin of error The Margin of Error can be calculated as follows: Margin of Error formula = (Z value)*(Standard Deviation)/( square root of sample size) The Z value is determined by the confidence level you are trying to achieve Confidence level, z* value (ref: http://people.richland.edu/james/lecture/m170/ch08-int.html) 90% 1.645 95% 1.96 98% 2.33 99% 2.58 --------------------------------------------------------------

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