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Chapter 7: Problem 74

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.

Short Answer

Expert verified
In terms of sample proportions, a 'large' sample is often considered one where np and nq are both greater than 10. This ensures that the sample proportion \(\hat{p}\) is approximately normally distributed, as predicted by the central limit theorem.

Step by step solution

01

Understanding Central Limit Theorem

The central limit theorem stipulates 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. This will hold true regardless of whether the source population is normal or skewed, provided the sample size is reasonably large (usually n > 30).
02

Application to Sample Proportions

In terms of proportions, the central limit theorem states that if \(p\) is the probability of success on a single trial and \(q=1-p\) (the probability of failure), and if n is the number of trials (sample size), then when n is large enough, the binomial distribution can be approximated by a normal distribution with \(\mu=np\) and \( \sigma=\sqrt{npq} \).
03

Defining a Large Sample

There is no hard and fast rule for what constitutes a 'large' sample. However, a common guideline is that if both np and nq are greater than 10, the sample is considered 'large'. This ensures that the sampling distribution of the proportion \(\hat{p}\) can be approximated by a normal distribution.

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Most popular questions from this chapter

A company manufactured six television sets on a given day, and these TV sets were inspected for being good or defective. The results of the inspection follow. Good Good \(\quad \begin{array}{llll}\text { Defective } & \text { Defective } & \text { Good } & \text { Good }\end{array}\) r. What proportion of these TV sets are good? b. How many total samples (without replacement) of size five can be selected from this population? c. List all the possible samples of size five that can be selected from this population and calculatc the sample proportion, \(\hat{p}\), of television sets that are good for each sample. Prepare the sampling distribution of \(\hat{p}\). d. For each sample listed in part c. calculate the sampling error.

A machine at Keats Corporation fills 64 -ounce detergent jugs. The probability distribution of the amount of detergent in these jugs is normal with a mean of 64 ounces and a standard deviation of \(.4\) ounce. The quality control inspector takes a sample of 16 jugs once a week and measures the amount of detergent in these jugs. If the mean of this sample is either less than \(63.75\) ounces or greater than \(64.25\) ounces, the inspector concludes that the machine needs an adjustment. What is the probability that based on a sample of 16 jugs, the inspector will conclude that the machine needs an adjustment when actually it does not?

If all possible samples of the same (large) size are selected from a population, what percentage of all the sample means will be within \(1.5\) standard deviations of the population mean?

The standard deviation of the 2009 gross sales of all corporations is known to be \(\$ 139.50\) million. Let \(\bar{x}\) be the mean of the 2009 gross sales of a sample of corporations. What sample size will produce the standard deviation of \(\bar{x}\) equal to \(\$ 15.50\) million?

Suppose the incomes of all people in the United States who own hybrid (gas and electric) automobiles are normally distributed with a mean of \(\$ 78,000\) and a standard deviation of \(\$ 8300\). Let \(\bar{x}\) be the mean income of a random sample of 50 such owners. Calculate the mean and standard deviation of \(\bar{x}\) and describe the shape of its sampling distribution.
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