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Answer to Question #6467 in Statistics and Probability for carol

Question #6467
A researcher is interested in whether students who attend private high schools have higher average SAT Scores than students in the general population. A random sample of 90 students at a private high school is tested and and a mean SAT score of 1030 is obtained. The average score for public high school student is 1000 (σ= 200).


What is the Z critical value (Z cv ) using a 0.05 alpha level?

Should H0 be rejected? What should the researcher conclude?

Determine the 95 % confidence interval for the population mean, based on the sample mean.
Expert's answer
Let Ho: average private high scools score (m)<=aver. general score (M).
So
we can use one-tailed z-test.
z-score equals
Z=(m-M)*sqrt(n)/σ=(1030-1000)*sqrt(90)/200=1.42
Z cv=1.645 (we take it from
normal distribution table for alpha=0.05)
Z<Z cv, so we can't reject Ho
and conclude that the observed data set provides no strong evidence against the
null hypothesis.
The 95% confidence interval based on the sample mean (now we
suppose that population mean is uknown with known standard deviation) is
m±Z*σ/sqrt(n), where Z*=Z cv for level alpha/2 (! because to get an interval we
need to use two-sided estimation). Z*=1.96 and 95% confidence interval is
(1030-41.32;1030+41.32)=(988.68,1071.32)

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