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Chapter 7: Problem 76

Indicate in which of the following cases the central limit theorem will apply to describe the sampling distribution of the sample proportion. a. \(n=20\) and \(p=.45\) b. \(n=75\) and \(p=.22\) c. \(n=350\) and \(p=.01\) d. \(n=200\) and \(p=.022\)

Short Answer

Expert verified
The central limit theorem can be applied for case a and case b, but it cannot be applied to case c and case d due to failure in satisfying both the conditions.

Step by step solution

01

Evaluate Case a

For case a, where \(n=20\) and \(p=.45\), evaluate \(n \cdot p\) and \(n \cdot (1-p)\). So, \(20 \cdot 0.45 = 9\) and \(20 \cdot (1-0.45) = 11\). Since both these values are greater than 5, the central limit theorem can be applied.
02

Evaluate Case b

For case b, where \(n=75\) and \(p=.22\), evaluate \(n \cdot p\) and \(n \cdot (1-p)\). So, \(75 \cdot 0.22 = 16.5\) and \(75 \cdot (1-0.22) = 58.5\). Since both these values are greater than 5, the central limit theorem can be applied.
03

Evaluate Case c

For case c, where \(n=350\) and \(p=.01\), evaluate \(n \cdot p\) and \(n \cdot (1-p)\). So, \(350 \cdot 0.01 = 3.5\) and \(350 \cdot (1-0.01) = 346.5\). Since \(n \cdot p\) is not greater than 5, the central limit theorem cannot be applied.
04

Evaluate Case d

For case d, where \(n=200\) and \(p=.022\), evaluate \(n \cdot p\) and \(n \cdot (1-p)\). So, \(200 \cdot 0.022 = 4.4\) and \(200 \cdot (1-0.022) = 195.6\). Since \(n \cdot p\) is not greater than 5, the central limit theorem cannot be applied.

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Most popular questions from this chapter

The delivery times for all food orders at a fast-food restaurant during the lunch hour are normally distributed with a mean of \(7.7\) minutes and a standard deviation of \(2.1\) minutes. I.et \(\bar{x}\) be the mean delivery time for a random sample of 16 orders at this restaurant. Calculate the mean and standard deviation of \(\bar{x}\), and describe the shape of its sampling distribution.

A chemist has a 10-gallon sample of river water taken just downstream from the outflow of a chemical plant. He is concerned about the concentration, \(c\) (in parts per million), of a certain toxic substance in the water. He wants to take several measurements, find the mean concentration of the toxic substance for this sample, and have a \(95 \%\) chance of being within \(.5\) part per million of the true mean value of \(c\). If the concentration of the toxic substance in all measurements is normally distributed with \(\sigma=.8\) part per million, how many measurements are necessary to achieve this goal?

According to the central limit theorem, the sampling distribution of \(\hat{p}\) is approximately normal when the sample is large. What is considered a large sample in the case of the proportion? Briefly explain.

When is an estimator said to be consistent? Is the sample mean, \(\bar{x}\), a consistent estimator of \(\mu\) ? Explain.

Let \(\bar{x}\) be the mean of a sample selected from a population. a. What is the mean of the sampling distribution of \(\bar{x}\) equal to? b. What is the standard deviation of the sampling distribution of \(\bar{x}\) equal to? Assume \(n / N \leq .05\).
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