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Statistics Problem Set: No-Hitters in the MLB

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4.200 Use a statistical software package to generate 100 random samples of size n = 2 from a population characterized be a uniform probability distribution (Section 4.9) which c = 0 and d = 10. Compute mean x for each sample and plot a frequency distribution for the 100 mean x values. Repeat this process for n = 5,10, 20 and 50. Explain how your plots illustrate the Central Limit Theorem.

4.202 A poll by the Gallup Organization sponsored by Philadelphia-based CIGNA Integrated Care found that about 40% of employees have missed work due to a musculoskeletal (back injury of some kind. Let x be the number of sampled workers who have missed work due to a back injury.
a. Explain why x is approximately a binomial random variable.
b. Use the Gallup poll data to estimate p for the binomial random variable of part a.
c. A random sample of 10 workers is to be drawn from a particular manufacturing plant. Use the p from part b to find the mean and standard deviation of x, the number of workers who missed work due to back injuries.
d. For the sample in part c, find the probability that exactly one worker missed work due to a back injury. That more than one worker missed work due to a back injury.

4.204 In baseball, a no-hitter is a regulation nine-inning game in which the pitcher yields no hits to the opposing batters. Chance reported on a study of no-hitters in Major League Baseball (MLB). The initial analysis focused on the total number of hits yielded per game per team for nine-inning MLB games. The distribution of hits/nine-innings is approximately normal with mean 8.72 and standard deviation 1.10.
a. What percentage of nine-inning MLB games result in fewer than six hits?
b. Demonstrate, statistically, why a no-hitter is considered an extremely rare occurrence in MLB.

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Q200. As the n increases, the mean of a sufficiently large number of independent random variables, each with finite mean and variance, will be approximately normally ...

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The solution discusses the no-hitters in the MLB in the statistics problem set.