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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

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Jul 10, 2011
Statistics – Margin of Error
by: Staff


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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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