Question

1. (6 pts) The Graduate Record Examination (GRE) is a test required for admission to many U.S. graduate schools. Students' scores on the quantitative portion of the GRE follow a normal distribution with mean 150 and standard deviation 8.8. In addition to other qualifications, a score of at least 160 is required for admission to a particular graduate school.

1. What proportion of combined scores can be expected to be over 160?

2. What proportion of combined GRE scores can be expected to be between 155 and 160?

3. What is the probability that a randomly selected student will score less than 150 points?

4. Historically, Department of History at NYU has admitted students whose quantitative GRE score is at least at the 61st percentile. What is the lowest GRE score of the students they admit?

5. Determine the range of scores that make up the middle 95% of scores.  

2.(4 pts) The Graduate Record Examination (GRE) is a test required for admission to many U.S. graduate schools. Students' scores on the quantitative portion of the GRE follow a normal distribution with mean 150 and standard deviation 8.8. In addition to other qualifications, a score of at least 160 is required for admission to a particular graduate school.

Suppose n = 16 randomly selected students take the GRE on the same day.

1. Describe the sampling distribution of the sample mean for the quantitative GRE Scores for the 16 students. (give the shape, mean, and SD)

2. What is the probability that a random sample of 16 students has a mean score on the quantitative portion of the GRE that is less than 147? Would this be an unusual outcome.

3. (4 pts) According to a recent article, 32% of drivers had driven drowsy in the past month. (Source: Thenationshealth.aphapublications.org/content/41/10/E52.fuII)

Suppose law enforcement officials are planning a survey of 1000 drivers to determine what proportion are driving drowsy.

1. Is whether a driver has driven drowsy in the past month qualitative or quantitative?

2. Would it be unusual if, in a random sample of 1000 drivers, 35% or more were driving drowsy in the last month? Include justifications for full credit.

3. What is the minimum number of drivers that must be sampled to be sure that the shape is approximately normal? Hint – use the formula

4. (6 pts) The Graduate Management Admission Test (GMAT) is a test required for admission into many masters of business administration (MBA) programs. Suppose a random sample of 8 students took the test, and their scores are given below.

699, 560, 414, 570, 521, 663, 727, 413

1. Find a point estimate of the population mean.

2. Construct a 95% confidence interval for the true mean score for the population.

3. Interpret the interval you constructed above.

4. How many students should be surveyed to estimate the mean score within 25 points with 95% confidence? Hint – use the formula

Answer 1

1)

P(x> 160)
= P(z > (160 - 150)/8.8)
= P(z > 1.1364)
= 0.1279

b)

P(155 < x < 160)
= P((155 - 150)/8.8 < z < ( 160 -150)/8.8)
= P(0.5682 < z < 1.1364)
= 0.1571

c)

P(x< 150)
= P(z < (150 - 150)/8.8)
= P(z < 0)
= 0.5

d)

z value at 61 percentile = 0.2793

z = (x -mean)/s

0.2793 = ( x - 150)/8.8

x = 152.4580

e)

z value at 95% 1.96
CI = mean +/- z *s
= 150 +/- 1.96 * 8.8
= (132.7520,167.2480)

Lower Bound = 132.7520
Upper bound = 167.2480