Answer :
Final answer:
To estimate the mean SAT math score with a 90% confidence level and a margin of error below 10, the researcher should consider a sample size of about 274 using the formula for sample size in a normal distribution and assuming the standard deviation to be approximately 100.
Explanation:
The subject of this question falls under Statistics, a branch of Mathematics. The researcher from the University of New Hampshire needs to determine the sample size required to estimate the mean SAT math score with a 90% confidence level and a margin of error below 10.
In a normal distribution, the standard deviation can be assumed to be approximately 100 (as per the Rule of Standard Deviations for SAT scores). To ensure that the margin of error is below 10, the researcher needs to use the formula for the sample size (n) in a normal distribution which is (Z*σ/E)^2.
For a 90% confidence level, the z-value (Z) is 1.645 (from the z-table or statistical software). Assuming σ (standard deviation) as 100 and E (margin of error) as 10, substitute these values into the formula:
n = (Z*σ/E)^2 = (1.645*100/10)^2
After calculating, we find that the result is approximately 274. Thus, the researcher would need a sample size of about 274 to ensure the margin of error is below 10 for the estimation of mean SAT math score of the incoming class with 90% confidence.
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