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Chapter 6: Problem 4

The proportion of a population with a characteristic of interest is \(p=0.37\). Find the mean and standard deviation of the sample proportion \(\hat{\boldsymbol{P}}\) obtained from random samples of size 125 .

Short Answer

Expert verified
The mean is 0.37, and the standard deviation is approximately 0.0432.

Step by step solution

01

Identify the Parameters

We start by identifying the given parameters in the problem. We have the population proportion \( p = 0.37 \) and the sample size \( n = 125 \).
02

Calculate Mean of Sample Proportion

The mean of the sample proportion \( \hat{P} \) is the same as the population proportion \( p \). Therefore, the mean is \( \mu_{\hat{P}} = p = 0.37 \).
03

Calculate Standard Deviation of Sample Proportion

The standard deviation of the sample proportion is given by the formula \( \sigma_{\hat{P}} = \sqrt{\frac{p(1-p)}{n}} \). Substitute the values to get \( \sigma_{\hat{P}} = \sqrt{\frac{0.37 \times 0.63}{125}} \).
04

Solve for Numerical Value

Calculate the expression: \( \sigma_{\hat{P}} = \sqrt{\frac{0.2331}{125}} = \sqrt{0.0018648} \approx 0.0432 \).

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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 is a key concept in statistics that helps us understand what fraction or percentage of the entire population possesses a particular characteristic of interest. Imagine you're studying a large group of people to find out how many of them own a pet. If 37% of the people in your group have pets, this number, 0.37, is the population proportion or simply, the parameter \( p \).

  • It's not usually feasible to know the exact proportion for a large population, so we use samples to estimate it.
  • The population proportion gives us a baseline or expectation when studying samples from the population.
Understanding the population proportion sets the stage for what we expect when drawing random samples from this population, which is crucial for making predictions and decisions based on sample data.
Mean of Sample Proportion
The mean of the sample proportion, represented as \( \mu_{\hat{P}} \), is intuitively straightforward. It is the average proportion of having that characteristic in all possible samples of a given size from the population. Notably, it happens to be equal to the actual population proportion, \( p \). That means if you were to take many samples and calculate the proportion of the characteristic in each, their average should closely resemble \( p = 0.37 \).

  • Example: if the population proportion is 37%, then the average proportion expected in your samples will also be 0.37.
  • This property lays the foundation for the inferential statistics since it ensures that our sampling distribution is unbiased.
This concept is helpful to confirm that the sample proportion remains a reliable estimate of the population parameter with each sample taken, given a large enough sample size.
Standard Deviation of Sample Proportion
The standard deviation of the sample proportion, \( \sigma_{\hat{P}} \), tells us how much variability can be expected in the sample proportions around their mean value, which is \( p \). Its formula is \( \sigma_{\hat{P}} = \sqrt{\frac{p(1-p)}{n}} \), where \( n \) is the sample size. This calculation reflects the spread of the sample proportions.

  • The standard deviation helps assess how accurately a given sample proportion estimates the population proportion.
  • In our exercise, with \( p = 0.37 \) and sample size \( n = 125 \), the standard deviation computes to approximately 0.0432. This quantifies the degree of sampling error to expect.
Higher sample sizes typically lead to smaller standard deviations, indicating more precision in estimating the population proportion. Understanding this concept is fundamental for gauging the reliability and accuracy of sample-based estimates.

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Most popular questions from this chapter

Suppose the mean weight of school children's bookbags is 17.4 pounds, with standard deviation 2.2 pounds. Find the probability that the mean weight of a sample of 30 bookbags will exceed 17 pounds.

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Suppose that in a certain region of the country the mean duration of first marriages that end in divorce is 7.8 years, standard deviation 1.2 years. Find the probability that in a sample of 75 divorces, the mean age of the marriages is at most 8 years.
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