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Chapter 8: Problem 24

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

Short Answer

Expert verified
When the population proportion is equal to 1, it means that every member of the population is a "success." Using the formula for the standard deviation of the sample proportion (\(SD(\hat{p}) = \sqrt{\frac{p(1-p)}{n}}\)), when \(p = 1\), the equation simplifies to \(SD(\hat{p}) = \sqrt{0}\), which results in \(SD(\hat{p}) = 0\). This indicates no variation among sample proportions, as every sample will also have a proportion of 1.

Step by step solution

01

Define \(\hat{p}\) and population proportion

First, let's define the terms we'll be using in this problem. The sample proportion \(\hat{p}\) is the proportion of successes in a sample from a population. The population proportion, on the other hand, represents the proportion of successes in the entire population.
02

Write the formula for the standard deviation of \(\hat{p}\)

The formula for the standard deviation of the sample proportion, denoted as \(SD(\hat{p})\), is given by: \[SD(\hat{p}) = \sqrt{\frac{p(1-p)}{n}}\] where \(p\) is the population proportion, \(n\) is the sample size, and \(SD(\hat{p})\) represents the amount of variation we expect to see among the sample proportions.
03

Set the population proportion (\(p\)) to 1

As the problem asks us to explain why the standard deviation of \(\hat{p}\) is equal to 0 when the population proportion is equal to 1, we need to set \(p = 1\) in the formula for \(SD(\hat{p})\). Doing this, we get: \[SD(\hat{p}) = \sqrt{\frac{1(1-1)}{n}}\]
04

Simplify the equation

Now let's simplify the equation. Since \(1 - 1 = 0\), we have: \[SD(\hat{p}) = \sqrt{\frac{1(0)}{n}}\] and \[SD(\hat{p}) = \sqrt{\frac{0}{n}}\] Because 0 divided by any number equals 0, the equation simplifies to: \[SD(\hat{p}) = \sqrt{0}\]
05

Conclude the explanation

Since the square root of 0 is 0, we can conclude that the standard deviation of \(\hat{p}\) is equal to 0 when the population proportion is equal to 1: \[SD(\hat{p}) = 0\] This makes intuitive sense because if the population proportion is equal to 1, every member of the population is a "success," resulting in no variation among sample proportions.

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Key Concepts

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

Population Proportion
In statistics, the population proportion is a fundamental concept that represents the fraction of the entire population that has a particular characteristic of interest. For example, if we are studying whether people prefer online learning to traditional learning, the population proportion would represent the percentage of people in the entire population who prefer online learning.
In our problem, the population proportion ( ), is assumed to be 1. This means that 100% of the population has the characteristic you're studying. When the population proportion equals 1, it implies there is no diversity or variation within the population on this trait. All individuals exhibit the same trait. Understanding the population proportion is important for statistical analyses, including estimating variances and calculating probabilities. It's important to have a clear grasp of this concept to further comprehend how sample proportions relate to the larger population context.
Sample Proportion
The sample proportion, denoted as , is the proportion of individuals in a sample that exhibit a particular characteristic. It acts as an estimate of the population proportion. In our example, if 70 out of 100 sampled people prefer online learning, the sample proportion is 0.7.

The sample proportion provides a practical way to gauge the overall population trait when you cannot survey everyone. It also plays a critical role in inferential statistics. For instance:
  • It helps infer the population characteristics from which the sample is drawn.
  • Acts as the foundation for confidence intervals and hypothesis testing.
When the population proportion is 1, every sample no matter its size will also have a sample proportion of 1, since every sample unit reflects the same population trait, ensuring zero variation amongst different samples.
Variation
Variation refers to how much sample results differ from each other and from the true population parameter. It's a measure of the spread or dispersion within a dataset. In our context, the standard deviation is used to quantify variation among sample proportions.
When a population proportion is 1, every individual in the population shares the same characteristic. Consequently, all samples also share this characteristic perfectly, leading to a sample proportion of 1 in each sample.

This total uniformity across samples results in a standard deviation of zero, as there is no discrepancy or variability amongst the samples. Thus:
  • In practical terms, variation among sample proportions provides insights into data reliability.
  • A higher variation indicates more spread in data, often leading to more robust statistical conclusions.
Zero variation implies perfect predictability across samples, where every sample is identical in terms of the population proportion it represents.

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Most popular questions from this chapter

The article "Younger Adults More Likely Than Their Elders to Prefer Reading the News" (October \(6,2016,\) www pewresearch.org/fact-tank/2016/10/06/younger- adults -more-likely-than-their-elders-to-prefer-reading-news/) estimated that only \(3 \%\) of those age 65 and older who prefer to watch the news, rather than to read or listen, watch the news online. This estimate was based on a survey of a large sample of adult Americans conducted by the Pew Research Center. Consider the population consisting of all adult Americans age 65 and older who prefer to watch the news and suppose that for this population the actual proportion who prefer to watch online is \(0.03 .\) a. A random sample of \(n=100\) people will be selected from this population and \(\hat{p},\) the proportion of people who prefer to watch online, 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 \(n=400\) 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 \(n=400 ?\) Explain.

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 \(\hat{p}=0.83 ?\)

Consider the following statement: The proportion of all calls made to a county \(9-1-1\) emergency number during the year 2017 that were nonemergency calls was 0.14 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, \(p=0.14\) or \(\hat{p}=0.14 ?\)

A random sample is to be selected from a population that has a proportion of successes \(p=0.25 .\) Determine the mean and standard deviation of the sampling distribution of \(\hat{p}\) for each of the following sample sizes: a. \(n=10\) d. \(n=50\) b. \(n=20\) e. \(n=100\) c. \(n=30\) f. \(n=200\)

The article "The Average American Is in Credit Card Debt, No Matter the Economy" (Money Magazine, February 9, 2016) reported that only \(35 \%\) of credit card users pay off their bill every month. Suppose that the reported percentage was based on a random sample of 1000 credit card users. Suppose you are interested in learning about the value of \(p,\) the proportion of all credit card users who pay off their bill every month. The following table is similar to the table that appears in Examples 8.4 and \(8.5,\) and is meant to summarize what you know about the sampling distribution of \(\hat{p}\) in the situation just described. The "What You Know" information has been provided. Complete the table by filling in the "How You Know It" column.
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