The scores of 12th-grade students on the National Assessment of Educational Prog

Question

The scores of 12th-grade students on the National Assessment of Educational Progress year 2000 mathematics test have a distribution that is approximately Normal with mean = 295 and standard deviation = 32.

a) Choose one 12th-grader at random. What is the probability (±0.1) that his or her score is higher than 295? _ Higher than 391 (±0.001)? _

b) Now choose an SRS of 4 twelfth-graders and calculate their mean score x. If you did this many times, what would be the mean of all the x-values? _

c) What would be the standard deviation (±0.1) of all the x-values?

d) What is the probability that the mean score for your SRS is higher
than 295? (±0.1)_ Higher than 391? (±0.0001)_

Explanation / Answer

a) The 12th grader has score of 295.

Find z score from given information and area corresponding to the z score. This gives the required probbaility.

z=(295-295)/32=0, the probability is 0.5.

z=(391-295)/32=3, the probability is 0.0014.

b) The mean score corresponding to 4 12th graders are: (280+267+250+275)/4=268

Per Central Limit Theorem the sample mean will be same as population mean that is 295.

c) Standarde deviation of the sampling distribution is : sigma/sqrt N=32/SQRT 100=3.2 (assume N is 100).

d) z1=(268-295)/32=-0.84 and z2=(268-391)/32=-3.84

Find areas corresponding to z scores.ans: P(z1>-0.84): 0.2995+0.5=0.7995 and P(Z2>-3.84): 0.9999

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