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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

The report "A Crisis in Civic Education" (January 2016, goacta.org/images/download/A_Crisis_in_Civic_Education .pdf, retrieved May 3,2017 ) indicated that in a survey of a random sample of 1000 recent college graduates, 96 indicated that they believed that Judith Sheindlin (also known on TV as "Judge Judy") was a member of the U.S. Supreme Court. Is it reasonable to conclude that the proportion of recent college graduates who have this incorrect belief is greater than 0.09 \((9 \%) ?\) (Hint: Use what you know about the sampling distribution of \(\hat{p}\). You might also refer to Example \(8.5 .)\)

For which of the following combinations of sample size and population proportion would the standard deviation of \(\hat{p}\) be smallest? $$ \begin{array}{ll} n=40 & p=0.3 \\ n=60 & p=0.4 \\ n=100 & p=0.5 \end{array} $$

Consider the following statement: Fifty people were selected at random from those attending a football game. The proportion of these 50 who made a food or beverage purchase while at the game was 0.83 . 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, \(p=0.83\) or \(\hat{p}=0.83 ?\)

The report "The Role of Two-Year Institutions in FourYear Success" (National Student Clearinghouse Research Center, \(2015,\) nscresearchcenter.org/wp- content/uploads /SnapshotReport17-2YearContributions.pdf, retrieved May 3 , 2017) states that nationwide, \(46 \%\) of students graduating with a four-year degree in the \(2013-2014\) academic year had been enrolled in a two-year college sometime in the previous 10 years. The proportion of students graduating with a fouryear degree in California with previous two-year college enrollment was estimated to be \(0.62(62 \%)\) for that year. Suppose that this estimate was based on a random sample of 1500 California four-year degree graduates. Is it reasonable to conclude that the proportion of California four-year degree graduates who attended a two-year college in the previous 10 years is different from the national figure? (Hint: Use what you know about the sampling distribution of \(\hat{p}\). You might also refer to Example \(8.5 .\) )

For which of the following sample sizes would the sampling distribution of \(\hat{p}\) be approximately normal when $$ \begin{array}{c} p=0.2 ? \text { When } p=0.8 ? \text { When } p=0.6 ? \\ n=10 \quad n=25 \\ n=50 \quad n=100 \end{array} $$
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