MATH1127: Introduction to Statistics (60022)
.. Lesson 7.7 Using the Central Limit Theorem (Combination)
Unit 3 Chapte
YUESLIVII $O$
In an effort to cut costs and improve profits, any US companies have been turning to outsourcing. In fact, according to Purchasing magazine, 54\% of companies surveyed outsourced some part of their manufacturing process in the past two to three years. Suppose 555 of these companies are contacted.
a. What is the probability percentage that 338 or more companies outsourced some part of their manufacturing process in the past two or three years? Round the percent to two decimal places.
b. What is the probability percendage that 285 or more companies outsourced some part of their manufacturing process in the past two or three years? Round the percent to two decimal places.
c. What is the probability percentage that $48 \%$ or less of these companies outsourced some part of their manufacturing process in the past two or three years? Round the percent to two decimal places.
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Step 1 ：The problem is asking for the probability that a certain number of companies outsourced some part of their manufacturing process. This is a binomial distribution problem, but since the number of trials (companies) is large, we can use the Central Limit Theorem to approximate the binomial distribution with a normal distribution.

Step 2 ：The Central Limit Theorem states that if you have a population with mean \(\mu\) and standard deviation \(\sigma\) and take sufficiently large random samples from the population with replacement, then the distribution of the sample means will be approximately normally distributed.

Step 3 ：In this case, the mean \(\mu\) is the expected number of companies that outsourced, which is the total number of companies times the probability of outsourcing (0.54). The standard deviation \(\sigma\) is the square root of the product of the total number of companies, the probability of outsourcing, and the probability of not outsourcing (1 - 0.54).

Step 4 ：We can then calculate the z-score for the number of companies in question (338 for the first question), which is the number of standard deviations that value is away from the mean. The z-score is calculated as \((X - \mu) / \sigma\), where X is the number of companies in question.

Step 5 ：Finally, we can use a z-table to find the probability that the z-score is greater than or equal to the calculated z-score. This is the probability that 338 or more companies outsourced.

Step 6 ：The probability that 338 or more companies outsourced some part of their manufacturing process in the past two to three years is 0.06%. This is a very small probability, which suggests that it is very unlikely that this many companies outsourced.

Step 7 ：Final Answer: The probability percentage that 338 or more companies outsourced some part of their manufacturing process in the past two to three years is \(\boxed{0.06\%}\).