Statistics bell curves

Question

3.  Scores on a mathematics exam given to ten-year-old children throughout Japan are Normally distributed with a mean of 520 and a standard deviation of 50. You might want to use the space directly below to sketch a bell-curve which may you find some of the answers.  

a) What proportion have a score between 420 and 620? a) _____________

b) What proportion have a score between 520 and 620? b) ____________

c) What proportion have a score between 470 and 620? c) _____________

d) What proportion have a score below 420? d) ____________

e) What proportion have a score between 420 and 820? e) ____________

f) What is the z-score of a child who has a test score of 570? f) _____________

g) What is the z-score of a child who has a test score of 600? g) _____________

h) What is the z-score of a child who has a test score of 463? h) ____________

i) A child who has an exam score at the 50th percentile has a score of i) _____________

j) A child who has an exam score at the 16th percentile has a score of j) _____________

Answer 1

mean = 520, standard deviation (SD) = 50

x = score

z = (x - mean)/SD

a) What proportion have a score between 420 and 620? a) _____________

P is probability. We transforms x to z using above formula.

P(420 < x < 620) = P( (420-520)/50 < z < (620-520)/50) = P(-2 < z < 2) = P(z < 2) - P(z < -2) = 0.9772 - 0.0228 = 0.9544

b) What proportion have a score between 520 and 620? b) ____________

P(520 < x < 620) = P( (520-520)/50 < z < (620-520)/50) = P(0 < z < 2) = P(z < 2) - P(z < 0) = 0.9772 - 0.5 = 0.4772

c) What proportion have a score between 470 and 620? c) _____________

P(470 < x < 620) = P( (470-520)/50 < z < (620-520)/50) = P(-1 < z < 2) = P(z < 2) - P(z < -1) = 0.9772 - 0.1587 = 0.8185

d) What proportion have a score below 420? d) ____________

P(x < 420) = P(z < (420-520)/50) = P(z < -2) = 0.0228

e) What proportion have a score between 420 and 820? e) ____________

P(420 < x < 820) = P( (420-520)/50 < z < (820-520)/50) = P(-2 < z < 6) = P(z < 6) - P(z < -2) = 1 - 0.0228 = 0.9772

f) What is the z-score of a child who has a test score of 570? f) _____________

z = (570 - 520)/50 = 1

g) What is the z-score of a child who has a test score of 600? g) _____________

z = (600 - 520)/50 = 1.6

h) What is the z-score of a child who has a test score of 463? h) ____________

z = (463-520)/50 = -1.14

i) A child who has an exam score at the 50th percentile has a score of i) _____________

50th percentile is the mean, so score is 520.

j) A child who has an exam score at the 16th percentile has a score of j) _____________

16th percentile corresponds to z-score of approximately -1

-1 = (x - 520)/50

-50 + 520 = x

470 = x