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Chapter 9: Problem 67

a. Does it seem reasonable that the mean of the sampling distribution of observed values of \(p^{\prime}\) should be \(p,\) the true proportion? Explain. b. Explain why \(p^{\prime}\) is an unbiased estimator for the population \(p\).

Short Answer

Expert verified
Yes, it is reasonable that the mean of the sampling distribution of observed values of p' should be p, the true proportion. This is in accordance with the law of large numbers. Additionally, p' is an unbiased estimator for p because the expected value of p' equals the population parameter p.

Step by step solution

01

Understanding the Mean of Sampling Distribution

The mean of the sampling distribution of a statistic is simply the expected value of the statistic. In other words, according to the law of large numbers, if we take a large number of samples and calculate the sample proportion (p') for each, the mean of all these sample proportions should be close to the true population proportion (p). So, it is indeed reasonable that the mean of the sampling distribution of observed values of p' should be p.
02

Understanding Unbiased Estimators

An unbiased estimator is a statistic that, on average, accurately measures the parameter it is estimating. Therefore, a sample proportion p' is an unbiased estimator for the population proportion p when the expected value of p' equals p. This is because if we were to sample many times, and each time calculate a new p', the average of all these p's would converge to the true p.
03

Conclusion

Now we see that p' as a good estimator because its expected value equals the population parameter, p, which is the property of an unbiased estimator. Also, the mean of the sampling distribution of observed values of p' should indeed be the true proportion p, in accordance with the law of large numbers.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Mean of Sampling Distribution
To understand the mean of the sampling distribution, let's break it down. Imagine you have a huge jar of marbles and you want to find out the proportion of red ones. Instead of counting each marble, you take some handfuls (samples) and note how many are red each time. This process gives you a kind of average red proportion after many samples. The average of all these proportions from your samples is what we call the mean of the sampling distribution.

The key idea here is that if your samples are truly random and you take enough of them, the mean of your sampling distribution will approach the true proportion of red marbles in the whole jar. This idea is supported by a principle called the law of large numbers. In essence, the more samples you take, the closer your average gets to the actual population proportion.
  • The sampling distribution's mean represents the central point.
  • It reflects the average outcome of all sample proportions.
  • It approximates the actual population proportion with enough samples.
Unbiased Estimator
An unbiased estimator is like aiming for the target and hitting the center exactly on average. It means that if you estimated over and over again, your estimates would average out to the hit the bullseye. In statistics, a sample proportion, represented often by \( p' \), is considered an unbiased estimator.

This is crucial because it means, on average, our sample proportion \( p' \) will equal the true population proportion \( p \). Simply put, when you calculate the average of your many samples, the result will zero in on the actual value you're trying to estimate.
  • An unbiased estimator's average value equals the parameter.
  • It ensures accurate predictions for population characteristics.
  • It offers a reliable way to infer from sample to population.
Population Proportion
The population proportion is a basic yet vital concept that tells us about the make-up of a population. Consider the earlier example with marbles. The population proportion could be the fraction of all the marbles that are red. It's important because it provides a summary of the entire population's characteristics in one single number.

Understanding the population proportion is crucial when conducting statistical analyses because it provides a benchmark to compare our sample proportions. If your samples are accurate and the sample size is large, your sample proportion should be close to the population proportion.
  • The population proportion summarizes population traits succinctly.
  • It's a benchmark for evaluating sample-derived estimates.
  • Accurate sample proportions should converge to this true proportion.

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Most popular questions from this chapter

A winemaker has placed a large order for the no. 9 corks described in Applied Example 6.13 (p. 285 ) and is concerned about the number of corks that might have smaller diameters. During the corking process, the corks are squeezed down to 16 to \(17 \mathrm{mm}\) in diameter for insertion into bottles with an \(18 \mathrm{mm}\) opening. The cork then expands to make the seal. The winemaker wants the corks to be as tight as possible and is therefore concerned about any that might be undersized. The diameter of each cork is measured in several places, and an average diameter is reported for each cork. The cork manufacturer has assured the winemaker that each cork has an average diameter within the specs and that all average diameters have a normal distribution with a mean of \(24.0 \mathrm{mm}\). a. Why does it make sense for the diameter of the cork to be assigned the average of several different diameter measurements? A random sample of 18 corks is taken from the batch to be shipped and the diameters (in millimeters) obtained: $$\begin{array}{llllllll}\hline 23.93 & 23.91 & 23.82 & 24.02 & 23.93 & 24.17 & 23.93 & 23.84 & 24.13 \\\24.01 & 23.83 & 23.74 & 23.73 & 24.10 & 23.86 & 23.90 & 24.32 & 23.83 \\\\\hline\end{array}$$ b. The average diameter spec is "24 \(\mathrm{mm}+0.6 \mathrm{mm} /\) \(-0.4 \mathrm{mm} . "\) Does it appear this order meets the spec on an individual cork basis? Explain. c. Does the sample in part a show sufficient reason to doubt the truthfulness of the claim, that the mean average diameter is \(24.0 \mathrm{mm},\) at the 0.02 level of significance? A different sample of 18 corks was randomly selected and the diameters (in millimeters) obtained: $$\begin{array}{lllllllll}\hline 23.90 & 23.98 & 24.28 & 24.22 & 24.07 & 23.87 & 24.05 & 24.06 & 23.82 \\\24.03 & 23.87 & 24.08 & 23.98 & 24.21 & 24.08 & 24.06 & 23.87 & 23.95 \\\\\hline\end{array}$$ d. Does the preceding sample show sufficient reason to doubt the truthfulness of the claim, that the mean average diameter is \(24.0 \mathrm{mm},\) at the 0.02 level of significance? e. What effect did the two different sample means have on the calculated test statistic in parts c and d? Explain. f. What effect did the two different sample standard deviations have on the calculated test statistic in parts c and d? Explain.

The May \(30,2008,\) online article "Live with Your Parents After Graduation?" quoted a 2007 survey conducted by Monster-TRAK.com. The survey found that \(48 \%\) of college students planned to live at home after graduation. How large of a sample size would you need to estimate the true proportion of students that plan to live at home after graduation to within \(2 \%\) with \(98 \%\) confidence?

A machine is considered to be operating in an acceptable manner if it produces \(0.5 \%\) or fewer defective parts. It is not performing in an acceptable manner if more than \(0.5 \%\) of its production is defective. The hypothesis \(H_{o}: p=0.005\) is tested against the hypothesis \(H_{a}: p>0.005\) by taking a random sample of 50 parts produced by the machine. The null hypothesis is rejected if two or more defective parts are found in the sample. Find the probability of the type I error.

Determine the critical region and critical value(s) that would be used to test the following using the classical approach: a. \(H_{o}: \sigma=0.5\) and \(H_{a}: \sigma>0.5,\) with \(n=18\) and \(\alpha=0.05\) b. \(H_{o}: \sigma^{2}=8.5\) and \(H_{a}: \sigma^{2}<8.5,\) with \(n=15\) and \(\alpha=0.01\) c. \(H_{o}: \sigma=20.3\) and \(H_{a}: \sigma \neq 20.3,\) with \(n=10\) and \(\alpha=0.10\) d. \(H_{o}: \sigma^{2}=0.05\) and \(H_{a}: \sigma^{2} \neq 0.05,\) with \(n=8\) and \(\alpha=0.02\) e. \(H_{o}: \sigma=0.5\) and \(H_{a}: \sigma<0.5,\) with \(n=12\) and \(\alpha=0.10\)

To test the hypothesis that the standard deviation on a standard test is \(12,\) a sample of 40 randomly selected students' exams was tested. The sample variance was found to be \(155 .\) Does this sample provide sufficient evidence to show that the standard deviation differs from 12 at the 0.05 level of significance?
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