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# Central Theorom Essay

671 words - 3 pages

Central Limit Theorem (Week 2)
QNT/561
January 5, 2011

Week 2 Assignment: Exercises 21, 22, and 34 (Ch. 8) of Statistical Techniques in Business & Economics:

#21 (a-c) a. The difference between the sample mean (bar X) and the population mean.
b. Yes.
c. The sample size is equal to the populations size

#22 (a-d) Reasons for sampling:
a. Time: Surveying and entire population can be time consuming. Sampling a small group can be quick and more efficient.

b. Cost: The cost to conduct a sampling will be more effective.

c. Accuracy: Small groups that are sampled can provide you with similar results in accuracy.

d. Destructive sampling -- for example, if you crash tested the entire population of cars for safety testing, you wouldn't have any left to drive -- so you only test a sample

#34 (a-e)

a. 5656.85

96
N = 36
sd = 0.03
mean = 3.01  (continued next page)
3.01 - 1.96*0.03/sqrt(36) to 3.01 + 1.96*0.03/sqrt(36)

3.0002 to 3.0198

#34 n = 50, m = 26, s = 6.2,

SE of the mean = s/sqrt n = 6.2/sqrt 50 = 0.8768, z = 1.96

Margin of error: E = Z * SE = 1.96 * 0.8768 = 1.7186

Lower Limit of the confidence interval = m - E = 24.2814

Upper Limit of the confidence interval = m + E = 26 + 1.7186 = 27.7186

Therefore, the confidence interval is 24.28 (lower limit) weeks and 27.72 (upper limit) weeks

28 weeks falls outside the above interval, it is not reasonable to assume that the population mean is 28 weeks.

#46 p = 14/220 = 0.06364, q = 1 - p = 0.9364
SE = sqrt(pq/n) = sqrt(0.06364 * 0.9364/220) = 0.0164
99% confidence, z = 2.576
**99% CI for the proportion of applicants who fail the test is given by
[p - z * SE, p + z * SE] = [0.06364 - 2.576 * 0.0164, 0.06364 + 2.576 * 0.0164]
= [0.0214, 0.106]
= [2.14%, 10.6%]
**Since 10% falls within the 99% CI but near the upper limit of 10.6%, it may not be reasonable to conclude that more than 10% are now failing the test.
(b) p = 14/400 = 0.035, q = 1 - p = 0.965
SE = sqrt(pq/n) = sqrt(0.035 * 0.965/400) = 0.0092
For 99% confidence, z = 2.576
The 99% CI for the proportion of employees who fail the test is given by
[p - z * SE, p + z * SE] = [0.035 - 2.576 * 0.0092, 0.035 + 2.576 * 0.0092]
= [0.0113, 0.0587]
= [1.13%, 5.87%]
**Since 5% falls within the 99% CI, we may conclude that less than 5 % are now failing
the test.

Discussion question 5 (at the end of Ch. 3) of Business Research Methods
#5 a. Determine the dilemma: Need to increase advertising sales to shoe manufacturers.
b. Determine the questions, examples below:
(1) Are men’s clothing stores an outlet for men’s shoes?
(2) Are shoe manufacturers a profitable sales source?
c. Determine the research questions:
(a) Do men’s clothing stores sell a large volume of shoes? (b) How much profit can be gained or is needed by the magazine from shoe manufacturers?

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