Question

1. A new test for reading aptitude is being rolled out in seventh graders nation-wide. The test is designed so that the average score in the population of test-takers is 65 with a standard deviation of 15. For this question, assume a sample size of 25 students randomly selected from this population. Show relevant z-scores.
   1. What proportion of such samples would we expect to have an average test score over 68?
   2. What proportion of such samples would we expect to have an average test score less than 70?
   3. What proportion of such samples would we expect to have an average test score over 64?

Answer 1

Solution :

Given that ,

mean = = 65

standard deviation = = 15

= / n = 15 / 25 = 3

a.

P( > 68) = 1 - P( < 68)

= 1 - P[( - ) / < (68 - 65) / 3]

= 1 - P(z < 1)

= 1 - 0.8413

= 0.1587

proportion = 0.1587

b.

P( < 70) = P(( - ) / < (70 - 65) / 3)

= P(z < 1.67)

= 0.9535

proportion = 0.9535

c.

P( > 64) = 1 - P( < 64)

= 1 - P[( - ) / < (64 - 65) / 3]

= 1 - P(z < -0.33)

= 1 - 0.3707

= 0.6293

proportion = 0.6293