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Chapter 6: Problem 8

In Exercises 6.7 and 6.8 , compute the standard error for sample proportions from a population with the given proportion using three different sample sizes. What effect does increasing the sample size have on the standard error? Using this information about the effect on the standard error, discuss the effect of increasing the sample size on the accuracy of using a sample proportion to estimate a population proportion. A population with proportion \(p=0.75\) for sample sizes of \(n=40, n=300,\) and \(n=1000 .\)

Short Answer

Expert verified
Upon computation, it's found that as the sample size increases, the standard error decreases. Therefore, an increase in sample size improves the accuracy of the sample proportion in estimating the population proportion.

Step by step solution

01

Calculate Standard Error for \(n=40\)

Substitute the given values \(p=0.75\) and \(n=40\) into the formula for standard error to get the value of SE: \(SE=\sqrt{0.75* (1-0.75)/40}\). Calculate to get the answer.
02

Calculate Standard Error for \(n=300\)

Similarly, substitute the given values \(p=0.75\) and \(n=300\) into the formula for standard error to get the value of SE: \(SE=\sqrt{0.75* (1-0.75)/300}\). Calculate to get the answer.
03

Calculate Standard Error for \(n=1000\)

Again, substitute the given values \(p=0.75\) and \(n=1000\) into the formula for standard error to get the value of SE: \(SE=\sqrt{0.75* (1-0.75)/1000}\). Calculate to get the answer.
04

Effect of increasing sample size on Standard Error

Upon computing the standard errors for the increasing sample sizes, it can be observed that as the sample size increases, the standard error decreases. This is because the standard error is inversely proportional to the square root of the sample size. So, as the sample size gets larger, the standard error gets smaller.
05

Discuss the effect on accuracy of estimation

The decrease in the standard error with an increase in sample size improves the accuracy of the sample proportion in estimating the population proportion. This is because a smaller standard error means that the sample proportion is expected to be closer to the population proportion.

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Key Concepts

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

Sample Proportion
The sample proportion is a statistical measure that helps us estimate the proportion of a specific characteristic in a population based on a sample drawn from that population. Imagine you're conducting a survey to find out how many people in a city like chocolate ice cream. If you interview 100 people and 75 of them say they like chocolate ice cream, the sample proportion is 0.75, or 75%. This number serves as an estimator of the entire population's preference.

The purpose of calculating a sample proportion is to gain an understanding of the likely values for the population proportion. The formula for the sample proportion is:
  • \( \text{Sample Proportion} = \frac{\text{Number of Successes}}{\text{Sample Size}} \)
This gives you a fraction representing the parts of your sample that have the characteristic you're investigating. The larger these parts, the higher the proportion will be, and vice versa.
Population Proportion
Population proportion is the actual proportion of individuals having a certain characteristic in the entire population. Let's say in the whole city, 0.75 or 75% of individuals like chocolate ice cream. That is the population proportion. However, often, it's hard to know the population proportion exactly, especially if the population is large.

In statistical studies, researchers aim to estimate this proportion using sample data because collecting data on the whole population is usually impractical. We use the sample proportion (as detailed earlier) to make this estimation. The closer your sample proportion to the population proportion, the more reliable your data is. Therefore, calculating a good estimate through increasing reliability is vital. To start, we assume the sample proportion is a good estimator of the population proportion.
Sample Size
Sample size is the number of observations or data points collected from a population for a study. It's a crucial factor since it can significantly impact the study's outcomes and reliability. For example, in the provided exercise, you have sample sizes of 40, 300, and 1,000. These numbers indicate how many individuals you're considering in your research, and the results will vary as the sample size changes.

One essential aspect of sample size is its effect on the standard error. The formula for standard error (SE) of a sample proportion is:
  • \( SE = \sqrt{ \frac{p(1 - p)}{n} } \)
Where \(p\) is the population proportion and \(n\) is the sample size. As you see from the formula, the sample size \(n\) is at the denominator under the square root, indicating that as \(n\) increases, the standard error decreases.

This means that larger sample sizes yield more precise estimates of the population proportion because they reduce the standard error, making your estimate closer to the true population proportion. This better accuracy is crucial in ensuring reliable and valid results in any statistical study.

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Most popular questions from this chapter

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If random samples of the given size are drawn from a population with the given mean and standard deviation, find the standard error of the distribution of sample means. Samples of size 1000 from a population with mean 28 and standard deviation 5

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