Question

the score on test for children in an eight grade class is normally distributed with a mean of 70 and a standard deviation of 4.2

what is the propability a test score is less than a 77?

what test score cuts off the bottom 25%?

what two test score cutoff the middle 65% of the distribution?

Answer 1

P(X < A) = P(Z < (A - mean)/standard deviation)

Mean = 70

Standard deviation = 4.2

P(a test score is less than a 77), P(X < 77) = P(Z < (77 - 70)/4.2)

= P(Z < 1.67)

= 0.9525

Let the test score that cutts off the bottom 25% be B

P(X < B) = 0.25

P(Z < (B - 70)/4.2) = 0.25

Take the value of Z corresponding to probability value of 0.2500 from cumulative standard normal distribution table (z table)

(B - 70)/4.2 = -0.67

B = 67.186

Let the two test scores that cut off the middle 65% of the distribution be M and N

P(X < M) = 0.5-0.65/2 = 0.175

P(Z < (M - 70)/4.2) = 0.175

Take the value of Z corresponding to probability value of 0.1750 from cumulative standard normal distribution table (z table)

(M - 70)/4.2 = -0.935

M = 66.073

P(X < N) = 0.5+0.65/2 = 0.825

P(Z < (N - 70)/4.2) = 0.825

Take the value of Z corresponding to probability value of 0.8250 from cumulative standard normal distribution table (z table)

(N - 70)/4.2 = 0.935

N = 73.927

The two test scores that cutoff the middle 65% of the distribution are 66.073 and 73.927.