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# Discussing the Main Points of the Central Limit Theorem for a Mean

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Question: Why is population shape of concern when estimating a mean? What does sample size have to do with it?

Scenario 1: A random sample of 10 miniature Tootsie Rolls was taken from a bag. Each piece was weighed on a very accurate scale. The results in grams were as given below:

3.087 3.131 3.241 3.241 3.270 3.353 3.400 3.411 3.437 3.477

(a) Construct a 90 percent confidence interval for the true mean weight.
(b) What sample size would be necessary to estimate the true weight with an error of ± 0.03 grams with 90 percent confidence?
(c) Discuss the factors which might cause variation in the weight of Tootsie Rolls during manufacture. (Data are from a project by MBA student Henry Scussel.)

Scenario 2: In 1992, the FAA conducted 86,991 pre-employment drug tests on job applicants who were to be engaged in safety and security-related jobs, and found that 1,143 were positive.
(a) Construct a 95 percent confidence interval for the population proportion of positive drug tests.
(b) Why is the normality assumption not a problem, despite the very small value of p? (Data are from Flying 120, no. 11 [November 1993], p. 31.)

##### Solution Summary

This solution discusses the outcome of the sample distribution as it approaches the normal distribution, how the spread of the distribution decreases, population shape and sample size, some factors that might cause variation, and the normality assumption. Additionally, the required calculations for the confidence interval and necessary sample size are provided. An Excel attachment file is attached which contains the solutions for Scenario 1 and Scenario 2.

##### Solution Preview

State the main points of the theorem:
(a) Point #1
The central limit theorem states that given a distribution with a mean μ and variance σ², the sampling distribution of the mean approaches a normal distribution with a mean (μ) and a variance σ²/N as N, the sample size, increases. The interesting thing about the central limit theorem is that no matter what the shape of the original ...

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