Question

1. The Graduate Management Admission Test (GMAT) is a test required for admission into many masters of business administration (MBA) programs. Total scores on the GMAT are normally distributed and historically had a population standard deviation of 113. The Graduate Management Admission Council (GMAC), who administers the test, claims that the mean total score is 529. Suppose a random sample of 8 students took the test and their scores are given below: 699, 560, 414, 570, 521, 663, 727, 413

a. Find a point estimate of the population mean.

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

c. Does this interval contain the value reported by GMAC?

d. How many students should be surveyed to estimate the mean score within 25 points with 99% confidence?

Answer 1

a)

point estimate = sample mean, xbar = 570.88

b)

sample standard deviation, σ = 113
sample size, n = 8

Given CI level is 95%, hence α = 1 - 0.95 = 0.05
α/2 = 0.05/2 = 0.025, Zc = Z(α/2) = 1.96

ME = zc * σ/sqrt(n)
ME = 1.96 * 113/sqrt(8)
ME = 78.31

CI = (xbar - Zc * s/sqrt(n) , xbar + Zc * s/sqrt(n))
CI = (570.88 - 1.96 * 113/sqrt(8) , 570.88 + 1.96 * 113/sqrt(8))
CI = (492.57 , 649.19)

c)

yes, it contains 529

d)

The following information is provided,
Significance Level, α = 0.01, Margin or Error, E = 25, σ = 113

The critical value for significance level, α = 0.01 is 2.58.

The following formula is used to compute the minimum sample size required to estimate the population mean μ within the required margin of error:
n >= (zc *σ/E)^2
n = (2.58 * 113/25)^2
n = 135.99

Therefore, the sample size needed to satisfy the condition n >= 135.99 and it must be an integer number, we conclude that the minimum required sample size is n = 136
Ans : Sample size, n = 136