(15.49 S-AQ) The total SAT scores of high school seniors in recent years have me

Question

(15.49 S-AQ) The total SAT scores of high school seniors in recent years have mean µ = 1013 and standard deviation s = 204. The distribution of SAT scores is roughly Normal. Give your answers as whole numbers. (a) Ramon scored 1070. If scores have a Normal distribution, what percentile of the distribution is this?

(a)(That is, what percent of scores are lower than Ramon's?) %

(b) Now consider the mean x ¯ ¯ ¯ x¯ of the scores of 50 randomly chosen students. If x ¯ ¯ ¯ x¯ = 1070, what percentile of the sampling distribution of x ¯ ¯ ¯ x¯ is this? % (Give your answer as XX.XX.)

Which of your calculations, (a) or (b), is less accurate because SAT scores do not have an exactly Normal distribution?

Explanation / Answer

Ans:

a)

z=(1070-1013)/204

z=0.28

P(z<0.28)=0.6100 or 61%

It will be 61st percentile

b)sampling distribution of sample means:

mean=1013

standard devation=204/sqrt(50)=28.85

z=(1070-1013)/28.85

z=1.976

P(z<1.976)=0.9759 or 97.59%

97.6th percentile

(a) is less accurate,as sampling distribution of sample means follows exact normal distribution,when sample size is sufficiently large.

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