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Chapter 7: Problem 112

Suppose you want to calculate \(P(a \leq \bar{x} \leq b)\), where \(a\) and \(b\) are two numbers and \(x\) has a distribution with mean \(\mu\) and standard deviation \(\sigma .\) If \(a<\mu

Short Answer

Expert verified
When the sample size gets larger, the probability \(P(a \leq \bar{x} \leq b)\) tends to increase towards 1.

Step by step solution

01

State What is Given

In this exercise, we are given the following parameters:\n\nThe variable \(x\) is distributed with mean \(\mu\) and standard deviation \(\sigma\).\nThe interval of interest is \([a,b]\) where \(a<\mu<b\).\n\nThe student is interested in determining what happens to the probability \(P(a \leq \bar{x} \leq b)\) as the sample size tends to infinity.
02

Apply the Central Limit Theorem

The Central Limit Theorem tells us, roughly speaking, that as the sample size tends to infinity, the distribution of \(x\) approaches a normal distribution with the same mean (\(\mu\)) and a standard deviation equal to the original standard deviation divided by the square root of the sample size \(\sigma / \sqrt{n}\). This is regardless of what the distribution of our variable was to begin with.
03

Determine What Happens to the Probability

As the sample size increases, the standard deviation of the sampling distribution of averages decreases, which means that the distribution is getting tighter and tighter around the mean. If \(a<\mu<b\), then as the distribution concentrates more and more around the mean, the probability \(P(a \leq \bar{x} \leq b)\) becomes closer and closer to 1, i.e., it increases towards 1.

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Most popular questions from this chapter

Let \(\hat{p}\) be the proportion of elements in a sample that possess a characteristic. \(\mathbf{a}_{\mathbf{+}}\) What is the mean of \(\hat{p}\) ? b. What is the standard deviation of \(\hat{p}\) ? Assume \(n / N \leq .05\). c. What condition(s) must hold true for the sampling distribution of \(\hat{p}\) to be approximately normal?

Briefly explain the meaning of a population distribution and a sampling distribution. Give an example of each.

For a population, \(N=30,000\) and \(p=.59\). Find the \(z\) value for each of the following for \(n=100\). a. \(p=.56\) b. \(\hat{p}=.68\) c. \(\hat{p}=.53\) d. \(\hat{p}=.65\)

Consider a large population with \(\mu=90\) and \(\sigma=18\). Assuming \(n / N \leq, 05\), find the mean and standard deviation of the sample mean, \(\bar{x}\), for a sample size of a. 10 b. 35

Explain the central limit theorem.
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