Question

Test Scores The mean score on a national aptitude test is 498 with a standard deviation of 100 points. (a) What percentage of the population has scores between 400 and 500\(?\) (b) If we sample 300 test-takers at random, about how many should have scores above 700\(?\)

Short answer

(a) The percentage of the population with test scores between 400 and 500 is approximately 34.45%. (b) We would expect about 7 out of 300 randomly sampled test-takers to have scores above 700.

Step-by-step solution

Calculating Z-scores

In a normal distribution, the Z-score for a value can be calculated using the formula: Z-value = (X-value – mean) / standard deviation. \n For X = 400: Z1 = \((400 - 498) / 100 = -0.98\) \n For X = 500: Z2 = \((500 - 498) / 100 = 0.02\)

Looking up probabilities in the Z-table

The Z-table gives us the probability that a score is below a certain Z-score. \n For Z1=-0.98, the probability P(Z<Z1) is 0.1635, which means 16.35% lie below the score of 400. \n For Z2=0.02, the probability P(Z<Z2) is 0.5080, or 50.80% lie below the score of 500. \n The percentage of scores between 400 and 500 will be P(400<X<500) = P(Z1<Z<Z2) = P(Z<Z2) - P(Z<Z1) = 0.5080 - 0.1635 = 0.3445 or 34.45%.

Calculating the Z-score for 700

For X = 700: Z3 = \((700 - 498) / 100 = 2.02\)

Probability for a score above 700

From the Z-table, the probability P(Z<2.02) = 0.9783 or 97.83% of the population score below 700. So, 1 - P(Z<2.02) = 1 - 0.9783 = 0.0217 or 2.17% would score above 700.

Calculate the expected count of test-takers with a score above 700

In a sample of 300 test-takers, the expected count of those scoring above 700 would be 300 * 0.0217 = 6.51. Since we cannot have a fraction of a person, we can round this to approx 7 people.