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Chapter 8: Problem 13

Consider the following statement: In a sample of 20 passengers selected from those who flew from Dallas to New York City in April 2012, the proportion who checked luggage was \(\underline{0.45}\). 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, \(p=0.45\) or \(\hat{p}=0.45 ?\)

Short Answer

Expert verified
a. The number 0.45 represents a SAMPLE proportion. b. The correct notation is \(\hat{p}=0.45\).

Step by step solution

01

Interpretation of the statement

In the statement, it is mentioned that a 'sample of 20 passengers' was taken out of those who traveled from Dallas to New York in April 2012. This implies that the number 0.45 is a calculation based on a subgroup or sample out of the total population of passengers. Hence, the proportion 0.45 represents a sample proportion.
02

Correct notation selection

Since the proportion represents a sample proportion, the correct notation to use is \(\hat{p}=0.45\). Here, \(\hat{p}\) denotes a sample proportion, as opposed to p which is generally used for a population proportion.

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Key Concepts

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

Sample Proportion
A sample proportion is a way to express what fraction or percentage of a sample exhibits a certain trait. In this case, the sample consists of 20 passengers who traveled from Dallas to New York City. Out of these passengers, 45% checked luggage, which our data represents as a sample proportion of 0.45.

Sample data helps us make inferences about a larger population without needing to examine every individual. This is useful in real-world scenarios where surveying or assessing an entire population could be challenging or impossible. In statistics, a sample proportion is often denoted by the symbol \( \hat{p} \).
  • It is calculated as \( \hat{p} = \frac{x}{n} \), where \( x \) is the number of successes (in this case, passengers who checked luggage) and \( n \) is the total number in the sample (20 passengers).
  • This makes \( \hat{p} = \frac{9}{20} = 0.45 \), if we assume 9 of the 20 passengers checked luggage.
Population Proportion
Understanding population proportion is vital when interpreting sample data. A population proportion is the fraction of a whole population that exhibits a particular characteristic. For example, if we were to evaluate all passengers flying from Dallas to New York City within that period, the calculation of how many passengers checked luggage would reflect the population proportion.

Population proportions are used to understand what percentage of the entire group behaves in a specific way, thus making them essential when comparing against sample data. This is important for validating results from sample data, providing a reference point.
  • In notation, population proportion is often represented as \( p \).
  • The major difference is that while \( \hat{p} \) applies to samples, \( p \) applies to the entire population.
Proportion Notation
The notation used to express proportions is crucial because it reflects whether the data concerns the sample or the population. This differentiation helps clarify what the numbers actually represent, and using incorrect notation can lead to misunderstandings.

When working with a sample proportion, we use \( \hat{p} \), which in our exercise is \( \hat{p} = 0.45 \). This indicates that we are working with data derived from a subgroup, not the entire population. Conversely, when discussing a population proportion, we use \( p \).
  • \( \hat{p} = 0.45 \) is used for interpreting the behavior of the sample.
  • \( p \) is reserved for data that is representative of the entire population.
The correct use of this notation is crucial in statistical analysis to avoid confusion.
Data Interpretation
Interpreting data correctly is an integral part of statistical analysis. When dealing with proportions, recognizing whether you are analyzing sample or population data can significantly alter the analysis and outcome.

In the given exercise, the number 0.45 corresponds to a sample proportion because it refers to a subset of passengers. Misinterpreting this could lead to the incorrect assumption that it represents every passenger from Dallas to New York City. Clear understanding of such distinctions ensures more accurate data representation and decision-making.
  • Data interpretation involves more than just acknowledging the numbers; it's about understanding the source and scope of the data.
  • Analyzing how data proportions like \( \hat{p} \) relate back to broader contexts can provide insights that affect policy decisions or business strategies.

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Most popular questions from this chapter

A random sample of 100 employees of a large company included 37 who had worked for the company for more than one year. For this sample, \(\hat{p}=\frac{37}{100}=0.37\). If a different random sample of 100 employees were selected, would you expect that \(\hat{p}\) for that sample would also be \(0.37 ?\) Explain why or why not.

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.

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

The report "California's Education Skills Gap: Modest Improvements Could Yield Big Gains" (Public Policy Institute of California, April \(16,2008,\) www.ppic.org) states that nationwide, \(61 \%\) of high school graduates go on to attend a two-year or four-year college the year after graduation. The proportion of high school graduates in California who go on to college was estimated to be \(0.55 .\) Suppose that this estimate was based on a random sample of 1,500 California high school graduates. Is it reasonable to conclude that the proportion of California high school graduates who attend college the year after graduation is different from the national figure? (Hint: Use what you know about the sampling distribution of \(\hat{p}\). You might also refer to Example \(8.5 .)\)

The report "New Study Shows Need for Americans to Focus on Securing Online Accounts and Backing Up Critical Data" (PRNewswire, October 29,2009 ) reported that only \(25 \%\) of Americans change computer passwords quarterly, in spite of a recommendation from the National Cyber Security Alliance that passwords be changed at least once every 90 days. For purposes of this exercise, assume that the \(25 \%\) figure is correct for the population of adult Americans. a. A random sample of size \(n=200\) will be selected from this population and \(\hat{p}\), the proportion who change passwords quarterly, 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 \(n=200 ?\) Explain. c. Suppose that the sample size is \(n=50\) rather than \(n=200 .\) 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 of 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 \(n=50 ?\) Explain.
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