Question

A large number of applicants for admission to graduate study in business are given an aptitude test. Scores are normally distributed with a mean of 460 and standard deviation of 80. What fraction of the applicants would you expect to have a score of 400 or above?

Answer 1

To find the fraction of applicants with a score of 400 or above, convert the score to a z-score and look it up in a standard normal distribution table. Subtract the resulting proportion from 1 to find the fraction above 400. Approximately 77.34% of the applicants would have a score of 400 or above.

Step-by-step explanation:

To find the fraction of applicants who would have a score of 400 or above, we need to find the area under the normal distribution curve to the right of 400. First, we need to convert the score of 400 to a z-score using the formula:

z = (x - μ) / σ

where z is the z-score, x is the score, μ is the mean, and σ is the standard deviation. In this case, the mean is 460 and the standard deviation is 80, so the z-score is:

z = (400 - 460) / 80 = -0.75

Once we have the z-score, we can look it up in a standard normal distribution table to find the proportion of the distribution that is below it. The table gives us a value of approximately 0.2266 for a z-score of -0.75. Since we want the fraction above 400, we can subtract this value from 1 to get:

1 - 0.2266 = 0.7734

Therefore, we would expect approximately 77.34% of the applicants to have a score of 400 or above.

Answer 2

The fraction or percentage of the applicants that we would expect to have a score of 400 or above is 77.34%

Step-by-step explanation:

Scores are normally distributed with a mean of 460 and a standard deviation of 80. For a value x, the associated z-score is computed as
`z = (x-460)/(80)`, therefore, the z-score for 400 is given by
`z_(0) = (400-460)/80 = -0.75`. To compute the fraction of the applicants that we would expect to have a score of 400 or above, we should compute the probability P(Z > -0.75) = 0.7734, i.e., the fraction or percentage of the applicants that we would expect to have a score of 400 or above is 77.34%