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Chapter 8: Problem 3

Consider the following statement: The Department of Motor Vehicles reports that the proportion of all vehicles registered in California that are imports is \(0.22 .\) 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, \(p=0.22\) or \(\hat{p}=0.22 ?\) (Hint: See definitions and notation on page \(403 .\) )

Short Answer

Expert verified
a. The number in boldface, 0.22, represents a population proportion. b. The correct notation is \(p = 0.22\).

Step by step solution

01

Determine the type of proportion

First, let's understand the difference between a sample proportion and a population proportion. A "sample proportion" refers to the proportion of a specific sample or portion from a population, while a "population proportion" refers to the proportion of the entire population. In this statement, the proportion refers to all vehicles registered in California - which indicates that it is a population proportion. a. Answer: The number in boldface, 0.22, represents a population proportion.
02

Identify the correct notation

When it comes to notation for proportions, "p" represents the population proportion and "p虃" (p-hat) represents the sample proportion. Since we determined in part a that the given proportion refers to a population proportion, the correct notation would be "p." b. Answer: The correct notation is \(p = 0.22\).

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Key Concepts

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

Sample Proportion
When we discuss sample proportions, we're talking about a slice of a whole pie. Imagine you're baking a pie, and you want to know what percentage is made of apples in a slice you cut. That slice represents your sample, and the apples in it are your sample proportion.

In the world of statistics, a sample proportion (\( \hat{p} \)) helps us understand a specific segment. It's not about the entire group but rather a particular subset.
  • It can change from sample to sample.
  • Provides a snapshot, not the full picture.
  • Varies as new samples are analyzed.
Whether you're dealing with cars, ages, or apples, distinguishing between sample proportion and its counterpart鈥攖he population proportion鈥攊s crucial for accurate statistical analysis.
Proportion Notation
In statistics, the way we write things down is just as important as the numbers themselves. Proportion notation keeps our math conversations clear. There are two primary notations: \( p \) and \( \hat{p} \).

Here's how to distinguish between the two:
  • Population Proportion (\( p \)): Refers to the whole group. Think of it like the entire pie, not just a piece. It remains the same unless the entire group changes.
  • Sample Proportion (\( \hat{p} \)): Refers to a part of the whole, like the piece of a pie. It is flexible and subject to variation, depending on the sample.
Using the right notation matters, as it conveys whether your focus is on a small sample or the broad population.
Statistical Definitions
Becoming comfortable with statistical definitions is like learning a new language. These definitions help form the foundation of easy communication in statistics.

A few key definitions to keep in mind include:
  • Population: The entire group you're interested in, like all vehicles in a state.
  • Sample: A smaller group selected from the population, for instance, only the vehicles in a particular city.
  • Proportion: A part or fraction of a whole, often represented in decimals or percentage terms.
  • Notation: The symbols we use, like \( p \) for population proportions, to reduce misunderstandings.
Benefiting from these definitions allows for effective analysis and conclusion in statistical tasks, making it easier to navigate quantitative data.

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Most popular questions from this chapter

Consider the two relative frequency histograms at the top of the next page. The histogram on the left was constructed by selecting 100 different random samples of size 40 from a population consisting of \(20 \%\) part-time students and \(80 \%\) full-time students. For each sample, the sample proportion of part-time students, \(\hat{p},\) was calculated. The \(100 \hat{p}\) values were used to construct the histogram. The histogram on the right was constructed in a similar way, but using samples of size 70 . a. Which of the two histograms indicates that the value of \(\hat{p}\) has smaller sample-to-sample variability? How can you tell? b. For which of the two sample sizes, \(n=40\) or \(n=70,\) do you think the value of \(\hat{p}\) would be less likely to be close to \(0.20 ?\) What about the given histograms supports your choice?

Suppose that a particular candidate for public office is favored by \(48 \%\) of all registered voters in the district. A polling organization will take a random sample of 500 of these voters and will use \(\hat{p},\) the sample proportion, to estimate \(p\). a. Show that \(\sigma_{\hat{p}}\), the standard deviation of \(\hat{p},\) is equal to 0.022 . b. If for a different sample size, \(\sigma_{j}=0.071,\) would you expect more or less sample-to-sample variability in the sample proportions than when \(n=500 ?\) c. Is the sample size that resulted in \(\sigma_{\hat{p}}=0.071\) larger than 500 or smaller than \(500 ?\) Explain your reasoning.

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

Some colleges now allow students to rent textbooks for a semester. Suppose that \(38 \%\) of all students enrolled at a particular college would rent textbooks if that option were available to them. If the campus bookstore uses a random sample of size 100 to estimate the proportion of students at the college who would rent textbooks, is it likely that this estimate would be within 0.05 of the actual population proportion? Use what you know about the sampling distribution of \(\hat{p}\) to support your answer.

The U.S. Census Bureau reported that in 2015 the proportion of adult Americans age 25 and older who have a bachelor's degree or higher is 0.325 ("Educational Attainment in the United States: 2015," www.census.gov, retrieved January 22,2017 ). Consider the population of all adult Americans age 25 and over in 2015 and define \(\hat{p}\) to the proportion of people in a random sample from this population who have a bachelor's degree or higher. a. Would \(\hat{p}\) based on a random sample of only 10 people from this population have a sampling distribution that is approximately normal? Explain why or why not. b. What are the mean and standard deviation of the sampling distribution of \(\hat{p}\) if the sample size is \(400 ?\) c. Suppose that the sample size is \(n=200\) rather than \(n=\) \(400 .\) 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.
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