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Chapter 8: Problem 27

A certain chromosome defect occurs in only 1 in 200 adult Caucasian males. A random sample of \(n=100\) adult Caucasian males is to be obtained. a. What is the mean value of the sample proportion \(\hat{p}\), and what is the standard deviation of the sample proportion? b. Does \(\hat{p}\) have approximately a normal distribution in this case? Explain. c. What is the smallest value of \(n\) for which the sampling distribution of \(\hat{p}\) is approximately normal?

Short Answer

Expert verified
The mean value of the sample proportion is 0.005 and the standard deviation of the sample proportion is 0.007. The sampling distribution of \(\hat{p}\) is not approximately normal in this case because \(n \cdot P < 5\). The smallest value of \(n\) that makes the sampling distribution of \(\hat{p}\) approximately normal is \(n = 1000\).

Step by step solution

01

Find the mean value of the sample proportion

Start by using the formula for the mean of the sample proportion which is the probability of the event. The given problem states that the chromosome defect occurs in 1 in 200 adult Caucasian males. So, the probability \(P\) that a randomly selected adult Caucasian male will have the chromosome defect is \(P = 1/200 = 0.005\). Therefore, \(\mu_{p} = P = 0.005\).
02

Compute the standard deviation of the sample proportion

Next, use the formula for the standard deviation of the sample proportion, \( \sigma_{p} = \sqrt{ \frac{P \cdot (1 - P)}{n}} \). Substitute \(P = 0.005\) and \(n = 100\) to get \(\sigma_{p} = \sqrt{ \frac{0.005 \cdot (1 - 0.005)}{100}} = 0.007\).
03

Determine if it follows an approximate normal distribution

Apply the Rule of Thumb for the central limit theorem which states that if the sample size is large enough, the distribution of the sample proportion will be approximately normal, regardless of the shape of the population distribution. According to this rule, the distribution of the sample proportion will be approximately normal if both \(n \cdot P \geq 5 \) and \(n \cdot (1 - P) \geq 5\) hold true. Substitute \(P = 0.005\) and \(n = 100\) to get \(n \cdot P = 100 \cdot 0.005 = 0.5\) and \(n \cdot (1 - P) = 100 \cdot (1 - 0.005) = 99.5\). While \(n \cdot (1 - P) \geq 5\) is true, \(n \cdot P \geq 5 \) is not, hence the distribution is not approximately normal.
04

Find the smallest value of \(n\)

To find the smallest value of \(n\), both conditions of the rule \(n \cdot P \geq 5 \) and \(n \cdot (1 - P) \geq 5\) must be true. If we consider \(n \cdot P \geq 5 \), we can solve for \(n\) by dividing both sides by \(P\). Thus, \(n \geq \frac{5}{P} = \frac{5}{0.005} = 1000\). So, the smallest value of \(n\) that makes the sampling distribution of \(\hat{p}\) approximately normal is \(n = 1000\).

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Most popular questions from this chapter

In the library on a university campus, there is a sign in the elevator that indicates a limit of 16 persons. In addition, there is a weight limit of 2500 pounds. Assume that the average weight of students, faculty, and staff on campus is 150 pounds, that the standard deviation is 27 pounds, and that the distribution of weights of individuals on campus is approximately normal. If a random sample of 16 persons from the campus is to be taken: a. What is the expected value of the distribution of the sample mean weight? b. What is the standard deviation of the sampling distribution of the sample mean weight? c. What average weights for a sample of 16 people will result in the total weight exceeding the weight limit of 2500 pounds? d. What is the chance that a random sample of 16 people will exceed the weight limit?

A random sample is selected from a population with mean \(\mu=100\) and standard deviation \(\sigma=10\). Determine the mean and standard deviation of the \(\bar{x}\) sampling distribution for each of the following sample sizes: a. \(n=9\) b. \(n=15\) c. \(n=36\) d. \(n=50\) e. \(n=100\) f. \(\quad n=400\)

Explain the difference between a population characteristic and a statistic.

A manufacturing process is designed to produce bolts with a \(0.5\) -inch diameter. Once each day, a random sample of 36 bolts is selected and the bolt diameters are recorded. If the resulting sample mean is less than \(0.49\) inches or greater than \(0.51\) inches, the process is shut down for adjustment. The standard deviation for diameter is \(0.02\) inches. What is the probability that the manufacturing line will be shut down unnecessarily? (Hint: Find the probability of observing an \(\bar{x}\) in the shutdown range when the true process mean really is \(0.5\) inches.)

Let \(x_{1}, x_{2}, \ldots, x_{100}\) denote the actual net weights (in pounds) of 100 randomly selected bags of fertilizer. Suppose that the weight of a randomly selected bag has a distribution with mean 50 pounds and variance 1 pound \(^{2}\). Let \(\bar{x}\) be the sample mean weight \((n=100)\). a. Describe the sampling distribution of \(\bar{x}\). b. What is the probability that the sample mean is between \(49.75\) pounds and \(50.25\) pounds? c. What is the probability that the sample mean is less than 50 pounds?
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