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Chapter 9: Problem 18

Consider taking a random sample from a population with \(p=0.40\) a. What is the standard error of \(\hat{p}\) for random samples of size \(100 ?\) b. Would the standard error of \(\hat{p}\) be larger for samples of size 100 or samples of size \(200 ?\) c. If the sample size were doubled from 100 to 200 , by what factor would the standard error of \(\hat{p}\) decrease?

Short Answer

Expert verified
a. The standard error of \(\hat{p}\) for random samples of size 100 is 0.049 b. The standard error of \(\hat{p}\) would be smaller for samples of size 200. c. The standard error of \(\hat{p}\) would decrease by a factor of \(\sqrt{2}\) when the sample size is doubled from 100 to 200.

Step by step solution

01

Calculating the Standard Error

To calculate the standard error of \(\hat{p}\) for random samples of size 100, the following formula is applied:\[SE=\sqrt{\frac{p(1-p)}{n}}\]where \(p\) is the population proportion and \(n\) is the sample size. Substituting \(p=0.40\) and \(n=100\), we get \[SE=\sqrt{\frac{0.40(1-0.40)}{100}}\]
02

Comparing Standard Errors

The standard error of \(\hat{p}\) would be smaller for larger samples \(n\). This is because the standard error formula has \(n\) (sample size) in the denominator. Hence, an increase in the sample size will decrease the standard error, thus the standard error for a sample size of 200 would be smaller than for a sample size of 100.
03

Determining the Factor of Decrease

To find the decrease factor of SE when the sample size is doubled (from 100 to 200), we consider that the standard error is inversely proportional to the square root of the sample size. So, if the sample size doubles, the standard error will decrease by a factor of \(\sqrt{2}\).

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Key Concepts

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

Population Proportion
The population proportion, denoted by the symbol \(p\), refers to the fraction of the total population that shares a particular characteristic. In statistical terms, it is the rate at which a particular outcome occurs in the population.
Calculated as a percentage, it is often expressed as a decimal between 0 and 1.
Here's how it might work:
  • If 40% of a city鈥檚 population supports a particular policy, then the population proportion \(p = 0.40\).
  • This means that out of every 10 people, 4 people express support for the policy.
Understanding population proportion is crucial because it serves as the basis for many statistical calculations and inferences.
We use it to estimate what percentage of the whole population might present the same characteristic or trend.
In this exercise, knowing \(p = 0.40\) helps us find other statistical values.
Sample Size
When collecting data for research, determining the right sample size, denoted as \(n\), is essential to ensure accurate results. The sample size is the number of observations or data points you include in your study.
Larger sample sizes generally yield more reliable estimates of the population characteristics.
For example, a sample size of
  • \(n = 100\) might give you a certain level of insight, but increasing it to \(n = 200\) could provide more accuracy.
The choice of sample size impacts the standard error, which is a measure of how much your sample proportion \(\hat{p}\) (the observed proportion) might vary from the population proportion \(p\). This exercise explores how changes in sample size influence the standard error, ultimately affecting conclusions drawn from the data.
Random Sampling
Random sampling is a fundamental concept in statistics where every member of a population has an equal chance of being chosen to be part of a sample.
This method is used to ensure that the sample accurately represents the population, minimizing biases.
The key advantages of random sampling are:
  • It provides a fair representation of the entire population.
  • It reduces selection biases.
  • It allows for the generalization of results from the sample to the population.
In the exercise, a random sample is assumed, which ensures that the calculation of the standard error is meaningful.
This assumption supports the idea that the sample proportion \(\hat{p}\) is an unbiased estimate of \(p\). Ensuring samples are random is a critical step in achieving valid and reliable results in statistical studies.
Standard Error Formula
The standard error (SE) helps in understanding the variability of a sample statistic, such as the sample proportion \(\hat{p}\).
The formula for calculating the standard error of \(\hat{p}\) is:\[SE=\sqrt{\frac{p(1-p)}{n}}\]In this formula:
  • \(p\) is the population proportion, representing the probability of a particular characteristic occurring.
  • \(n\) is the sample size, indicating the number of observations included.
The standard error decreases when the sample size increases. This is because the division by \(n\) in the formula means a larger denominator results in a smaller quotient.
In simpler terms, a larger sample size leads to more precise estimates, reducing the degree of error.
Understanding this formula allows us to predict how \(\hat{p}\) might differ from \(p\), offering insight into the reliability of our sample estimates.

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Most popular questions from this chapter

A car manufacturer is interested in learning about the proportion of people purchasing one of its cars who plan to purchase another car of this brand in the future. A random sample of 400 of these people included 267 who said they would purchase this brand again. For each of the three statements below, indicate if the statement is correct or incorrect. If the statement is incorrect, explain what makes it incorrect. Statement 1 : The estimate \(\hat{p}=0.668\) will never differ from the value of the actual population proportion by more than \(0.0462 .\) Statement 2 : It is unlikely that the estimate \(\hat{p}=0.668\) differs from the value of the actual population proportion by more than 0.0235 . Statement 3: It is unlikely that the estimate \(\hat{p}=0.668\) differs from the value of the actual population proportion by more than 0.0462 .

The Gallup Organization conducts an annual survey on crime. It was reported that \(25 \%\) of all households experienced some sort of crime during the past year. This estimate was based on a sample of 1,002 randomly selected adults. The report states, "One can say with \(95 \%\) confidence that the margin of sampling error is 卤3 percentage points." Explain how this statement can be justified.

A researcher wants to estimate the proportion of property owners who would pay their property taxes one month early if given a \(\$ 50\) reduction in their tax bill. Would the standard error of the sample proportion \(\hat{p}\) be larger if the actual population proportion were \(p=0.2\) or if it were \(p=0.4 ?\)

A researcher wants to estimate the proportion of students enrolled at a university who are registered to vote. Would the standard error of the sample proportion \(\hat{p}\) be larger if the actual population proportion was \(p=0.4\) or \(p=0.8\) ?

For estimating a population characteristic, why is an unbiased statistic with a small standard error preferred over an unbiased statistic with a larger standard error?
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