The scores on a national test for high school freshman have anormal distribution

Question

The scores on a national test for high school freshman have anormal distribution with a mean of 210 and a standard deviation of28.
(a) One tenth grader is randomly selected; what is the chance thathe/she scored over 220?
(b) A random sample of 40 tenth graders are selected. What is thechance that this group will average over 220? (Should this besmaller, larger, or approximately equal to (a)?)
(c) A larger sample of 100 tenth graders is selected. What is theprobability that this group will average over 220? (Should this besmaller, larger, or approximately equal to (a)?)

Explanation / Answer

a. Convert to normal score, then use standard normal table. (220-210)/28=.357 --> p(x>.357)=.3605 b. It should be smaller because it's a sample, reducing thestandard deviation. Convert to z. z=(220-210)/(28/40)=2.259 -->p(x>2.259)=.0119 c. It should be smaller because it's an even bigger sample,reducing the standard deviation. Convert to z.z=(220-210)/(28/100) --> z=3.571 -->p(z>3.571)=.000178

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