# Exercises from E-Text Week 5 Res341

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1 Week 5 Exercises from the e-text 6/30/2012 5. State the main points of the Central Limit Theorem for a mean. The central limit theorem allows for the approximation of the shape of the sample. The central limit theorem is the mean of a sufficiently large number of independent random variables that will be approximately normally distributed. It also describes the characteristics of the population of the means which has been created from the means of an infinite number of random population samples. 6. Why is the population shape of concern when estimating a mean? What does sample size have to do with it? Having a bell-shaped curve means it is normally distributed, and the central limit theorem does a good job at estimating the mean when you are dealing with a lot of variables. Sample size has everything to do with the central limit theorem, because the more samples you have the more the mean is going to be easily predicted. The more samples you have the more the central limit theorem is coming into play. 8.46 A random sample of 10 miniature tootsie rolls was taken from a bag. Each piece weighed on a very accurate scale. The results in grams were: 3.087 3.131 3.241 3.241 3.270 3.353 3.400 3.411 3.437 3.477 a. Construct a 90 % confidence interval for the true mean weight. b. What sample size would be necessary to estimate the true weight with an error of + or – 0.03 grams with a 90% confidence. c. Discuss the factors which might cause variation in the weight of Tootsie Rolls during manufacture. Sample mean: 3.3048 Standard deviation: .1320 90 percent confidence interval (taken from text): 1.812 a. X ± z σ√n 3.3048 ± .1320√10 = .0364 3.3048-.0364 = 3.2684 3.2684 to 3.3412 3.3048+.0364 = 3.3412 b. n =( zσE) ^ 2 (1.1812*.1320) / .03 ^2 = 63.57 round to 64