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Chapter 8: Problem 44

Consider the following statement: An inspector selected 20 eggs at random from the eggs processed at a large egg production facility. These 20 eggs were tested for salmonella, and the proportion of the eggs that tested positive for salmonella was \(\mathbf{0} . \mathbf{0} \mathbf{5}\). a. Is the number that appears in boldface in this statement a sample proportion or a population proportion? b. Which of the following use of notation is correct, $$ p=0.05 \text { or } p=0.05 ? $$

Short Answer

Expert verified
a. The number 0.05 is a sample proportion. b. The correct notation is \(\hat{p}=0.05\).

Step by step solution

01

Distinguish sample proportion from population proportion

In statistics, a sample is a subset of a population. The figure given, 0.05, represents the proportion of eggs that tested positive for salmonella out of the ones randomly selected, which is a sub-group of the whole population of eggs in the facility. Therefore, the aforementioned figure is a sample proportion.
02

Choose the correct notation

The symbol \(p\) is used to denote the population proportion, and the symbol \(\hat{p}\) is used to denote the sample proportion. Since 0.05 here represents a sample proportion, the correct notation is \(\hat{p}=0.05\).

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Key Concepts

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

Population Proportion
The concept of population proportion addresses the ratio or percentage of members in a whole group or population that have a particular attribute. Unlike a sample which is a smaller group chosen from the population, the population proportion reflects the reality of the entire group without exclusions. For instance, if we consider all eggs in a large egg production facility, and we aim to find out the proportion that is affected by salmonella, the calculation resulting from assessing every single egg would give us the population proportion.

Understanding this is critical, because it impacts decision making on a large scale. In health, economics, political science, and even in business settings, knowing the exact proportion in a whole population can guide strategies and policies. Incorrect assumptions about the population proportion could lead to significant errors in planning and resource allocation.
Statistical Notation
Using proper statistical notation is paramount in clear communication of mathematical concepts. Statistical notation allows statisticians and students alike to express complex ideas succinctly and accurately. For example, in statistics, the population proportion is traditionally represented by the symbol \(p\), while the sample proportion is annotated as \(\hat{p}\).

Hence, when you come across an equation or a statement, the notation used reveals a lot about what is being referenced. Is it discussing a population-wide measure or merely a subset? The distinction is crucial, as confusing one for the other may lead to unsound conclusions. Understanding and utilizing the correct statistical notation is not just a matter of academic correctness but serves as the bedrock for meaningful analysis and interpretation of data.
Probability Sampling
The technique of probability sampling refers to choosing a sample from a population in such a way that each member of the population has a known and likely non-zero chance of being included in the sample. This approach is vital for representativeness, ensuring that the sample accurately reflects the diversity of the population.

Common types of probability sampling include simple random sampling, stratified sampling, and cluster sampling, among others. Simple random sampling, used by the inspector to select the eggs for salmonella testing, gives each egg an equal chance to be chosen, thus minimizing biases. Properly conducted probability sampling leads to the possibility of generalizing the results of the sample to the population at large, which is the foundation of statistical inference.

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Most popular questions from this chapter

Suppose that the actual proportion of students at a particular college who use public transportation to travel to campus is \(0.15 .\) In a study of parking needs at the campus, college administrators would like to estimate this proportion. They plan to take a random sample of 75 students and use the sample proportion who use public transportation, \(\hat{p},\) as an estimate of the population proportion. a. Show that the standard deviation of \(\hat{p}\) is equal to \(\sigma_{p}=0.0412\) b. If for a different sample size, \(\sigma_{p}=0.0319,\) would you expect more or less sample-to-sample variability in the sample proportions than for when \(n=75 ?\) c. Is the sample size that resulted in \(\sigma_{p}=0.0319\) larger than 75 or smaller than \(75 ?\) Explain your reasoning.

The article "Unmarried Couples More Likely to Be Interracial" (San Luis Obispo Tribune, March 13,2002 ) reported that \(7 \%\) of married couples in the United States are mixed racially or ethnically. Consider the population consisting of all married couples in the United States. a. A random sample of \(n=100\) couples will be selected from this population and \(\hat{p},\) the proportion of couples that are mixed racially or ethnically, will be calculated. What are the mean and standard deviation of the sampling distribution of \(\hat{p} ?\) b. Is the sampling distribution of \(\hat{p}\) approximately normal for random samples of size \(n=100 ?\) Explain. c. Suppose that the sample size is \(n=200\) rather than \(n=\) \(100 .\) Does the change in sample size affect the mean and standard deviation of the sampling distribution of \(\hat{p} ?\) If so, what are the new values for the mean and standard deviation? If not, explain why not. d. Is the sampling distribution of \(\hat{p}\) approximately normal for random samples of size \(n=200 ?\) Explain.

The article referenced in the previous exercise also reported that \(38 \%\) of the 1,200 social network users surveyed said it was OK to ignore a coworker's friend request. If \(p=0.38\) is used as an estimate of the proportion of all social network users who believe this, is it likely that this estimate is within 0.05 of the actual population proportion? Use what you know about the sampling distribution of \(\hat{p}\) to support your answer.

Explain what the term sampling variability means in the context of using a sample proportion to estimate a population proportion.

Explain why the standard deviation of \(p\) is equal to 0 when the population proportion is equal to 1 .
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