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Chapter 6: Problem 58

Major League Baseball rules require that the balls used in baseball games must have circumferences between 9 and \(9.25\) inches. Suppose the balls produced by the factory that supplies balls to Major League Baseball have circumferences normally distributed with a mean of \(9.125\) inches and a standard deviation of \(.06\) inch. What percentage of these baseballs fail to meet the circumference requirement?

Short Answer

Expert verified
The percentage of baseballs that do not meet the circumference statistics required by the Major League Baseball rules can be calculated using the given step-by-step method.

Step by step solution

01

Find Z-scores

First, calculate the Z-scores for the lower limit (9 inches) and the upper limit (9.25 inches). The Z-score is calculated as \[ Z = \frac{X - \mu}{\sigma} \]. Substituting the given values into the Z-score formula, for the lower limit (\( Z_1 \)), \( X = 9 \) and for the upper limit (\( Z_2 \)), \( X = 9.25 \). Calculate those two Z-scores.
02

Calculate the probabilities

After getting the Z-scores, you have to calculate the proportion of data that lies between these Z-scores using a standard normal distribution table or a programming solution. Use the cumulative distribution function (CDF) to find these probabilities (\( P_1 \) and \( P_2 \)) where \( P_1 = P(Z \leq Z_1) \) and \( P_2 = P(Z \leq Z_2) \).
03

Find the percentage of balls outside the acceptable range

Subtract \( P_1 \) from \( P_2 \) to get the proportion of baseballs that fall within the acceptable range. Let's denote this result as \( P_r \). The proportion of baseballs that do not meet the required circumference will thus be \( 1 - P_r \). Multiply by 100 to get the percentage.

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Most popular questions from this chapter

A company that has a large number of supermarket grocery stores claims that customers who pay by personal checks spend an average of $$\$ 87$$ on groceries at these stores with a standard deviation of $$\$ 22 .$$ Assume that the expenses incurred on groceries by all such customers at these stores are normally distributed. a. Find the probability that a randomly selected customer who pays by check spends more than $$\$ 114$$ on groceries. b. What percentage of customers paying by check spend between $$\$ 40$$ and $$\$ 60$$ on groceries? c. What percentage of customers paying by check spend between $$\$ 70$$ and $$\$ 105 ?$$ d. Is it possible for a customer paying by check to spend more than $$\$ 185$$ ? Explain.

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What is the difference between the probability distribution of a discrete random variable and that of a continuous random variable? Explain.

For a binomial probability distribution, \(n=25\) and \(p=.40\). a. Find the probability \(P(8 \leq x \leq 13)\) by using the table of binomial probabilities (Table I of Appendix B). b. Find the probability \(P(8 \leq x \leq 13)\) by using the normal distribution as an approximation to the binomial distribution. What is the difference between this approximation and the exact probability calculated in part a?
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