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 have 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 |
514 |
570 |
521 |
663 |
727 |
513 |
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Find a point estimate of the population mean.
Sum of the sample of 8 students is = 4767
Divide by the number of the sample = 8
Answer = 595.875
Construct a 95% confidence interval (C.I.) for the true mean score for the population.
Looking for the 95% confidence level
Therefore, = x̅ +- Z95% (σ /sqrt(n) =595.875 +1.96*(113/SQRT(8))
= 674.18
=595.875 - 1.96*(113/SQRT (8))
= 517.57
Does this interval contain the value reported by GMAC?
Yes the interval contains the value that was prior given by the GMAC.
Now use this same sample to construct a 99% C.I. What is it? How does it compare to the 95% C.I.?
= x̅ +- Z99% (σ /sqrt(n) = 595.875 +2.58*(113/SQRT(8))
=698.95
= x̅ +- Z99% (σ /sqrt(n) = 595.875 - 2.58*(113/SQRT(8))
=492.8
= 492.8, 698.95
The interval is way much wider than it is with the CI of 95% thus we can say that the accuracy is 99% sure of the results acquired as compared to the 95% CI interval.
How many students should be surveyed to estimate the mean score within 25 points with 98% confidence?
25 = Z98% *(σ/sqrt(n)
25 = 2.33 * (113/√n))
n = (2.33*113/25) ^2
n = 110.9146
= 111 students
Therefore, it is evident that 111 students should be surveyed so as to estimate a mean score with the range of 25 points at a confidence of 98% confidence interval.
How many students should be surveyed to estimate the mean score within 25 points with 90% confidence?
25 = Z90% *(σ/sqrt(n)
25 = 1.645 * (113/√n))
n = (1.645*113/25) ^2
n = 55.28517
n = 55 students
Therefore, it is evident that 55 students should be surveyed so as to estimate a mean score with the range of 25 points at a confidence of 95% confidence interval.
Compare your answers from questions 5 and 6. What effect does decreasing the confidence level have on the sample size required?
Looking at the confidence level as it rises upwards so does the n value. Looking at question 5 above we can see that when the confidence interval was at 98% the number of n also was also very high at 111 students. In the same way when we now moved to question 6 and reduced the very same confidence interval to about 90% the number of n value also decreased to about only 55 students. Casting a wider range of confidence interval assures more accuracy.
