Question

(15.48 S-AQ) 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 µ = 322 and standard deviation s = 32.

Choose one 12th-grader at random. What is the probability (±±0.1) that his or her score is higher than 322?            Higher than 386 (±±0.001)?    

Now choose an SRS of 16 twelfth-graders and calculate their mean score x???x¯. If you did this many times, what would be the mean of all the x???x¯-values?    

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

What is the probability that the mean score for your SRS is higher
than 322? (±±0.1)      Higher than 386? (±±0.0001)

Answer 1

a) P(X > 322)

= P((X - )/ > (322 - )/)

= P(Z > (322 - 322)/32)

= P(Z > 0)

= 1 - P(Z < 0)

= 1 - 0.5

= 0.5

b) P(X > 386)

= P((X - )/ > (386 - )/)

= P(Z > (386 - 322)/32)

= P(Z > 2)

= 1 - P(Z < 2)

= 1 - 0.9772

= 0.023

c) For n = 16, mean = = 322

if we did this many times , the mean of all x = 322

standard deviation() = = 32/sqrt(16) = 8

d) P( > 322)

= P(( - )/() > (322 - )/())

= P(Z > (322 - 322)/8)

= P(Z > 0)

= 1 - P(Z < 0)

= 1 - 0.5

= 0.5

P( > 386)

= P(( - )/() > (386 - )/())

= P(Z > (386 - 322)/8)

= P(Z > 8)

= 1 - P(Z < 8)

= 1 - 1 = 0.0000