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Chapter 9: Problem 80

Will \(\hat{p}\) from a random sample of size 400 tend to be closer to the actual value of the population proportion when \(p=0.4\) or when \(p=0.7 ?\) Provide an explanation for your choice.

Short Answer

Expert verified
The sample proportion \(\hat{p}\) will tend to be closer to the actual value of the population proportion when \(p=0.7\). This is because the variance of the sample proportion is smaller when \(p=0.7\) (\(\sigma_{\hat{p}}^2 \approx 0.000525\)) compared to when \(p=0.4\) (\(\sigma_{\hat{p}}^2 \approx 0.0006\)), indicating that the values of \(\hat{p}\) are more concentrated around the true population proportion when \(p=0.7\).

Step by step solution

01

Understand the concept of sampling variation

The sample proportion is an estimate of the population proportion. Due to sampling variation, the sample proportion \(\hat{p}\) will vary from sample to sample. The variance of the sample proportion, which is a measure of the dispersion, can help us understand how close \(\hat{p}\) tends to be to the actual value of the population proportion \(p\).
02

Calculate the variance for both proportions

The formula for the variance of the sample proportion is given by \(\sigma_{\hat{p}}^2 = \dfrac{p(1 - p)}{n}\), where \(n\) is the sample size (400, in this case). For \(p=0.4\), we can calculate the variance as follows: \(\sigma_{\hat{p}}^2 = \dfrac{0.4(1 - 0.4)}{400}\), and for \(p=0.7\), the variance can be calculated as follows: \(\sigma_{\hat{p}}^2 = \dfrac{0.7(1 - 0.7)}{400}\).
03

Compare the variances

Calculate the variances for both proportions: For \(p=0.4\): \(\sigma_{\hat{p}}^2 = \dfrac{0.4(0.6)}{400} = \dfrac{0.24}{400} \approx 0.0006\). For \(p=0.7\): \(\sigma_{\hat{p}}^2 = \dfrac{0.7(0.3)}{400} = \dfrac{0.21}{400} \approx 0.000525\). As we see, the variance of the sample proportion is smaller when \(p=0.7\) than when \(p=0.4\).
04

Conclusion

Since the variance of the sample proportion is smaller when \(p = 0.7\) compared to when \(p=0.4\), we can conclude that \(\hat{p}\) from a random sample of size 400 will tend to be closer to the actual value of the population proportion when \(p = 0.7\). This is because a smaller variance indicates less dispersion, which means the values of \(\hat{p}\) are more concentrated around the true population proportion when \(p=0.7\).

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Key Concepts

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

Sampling Variation
When you pick a random sample from a population and calculate a statistic, such as a proportion, it's unlikely that it will be exactly the same as the population's true proportion. This discrepancy is what statisticians call sampling variation. It is a natural and expected behavior in statistics.

The variance of the sample proportion is a mathematical representation of sampling variation and is crucial to understanding how much we can expect our estimate \( \hat{p} \) to differ from the true population proportion \( p \) from one sample to another. One might assume that larger samples would have more variation, but the opposite is true: larger sample sizes actually lead to smaller variances, making the sample proportion a more reliable estimate of the population proportion.
Sample Size
The sample size, denoted as \( n \), plays an integral role in determining the accuracy and reliability of the sample proportion in estimating the population proportion. The variance of the sample proportion \( \sigma_{\hat{p}}^2 \) is inversely related to the sample size鈥搈eaning, as the sample size increases, the variance of the sample proportion reduces.

This is because, with more data, the statistic that you compute from the sample is based on more information and thus, is likely to be a better representation of the population. In the exercise solution presented, a size of 400 is used to minimize the variance and obtain a closer estimate \( \hat{p} \) of the actual population proportion \( p \).
Population Proportion Estimation
Estimating the population proportion is a fundamental task in statistics. The sample proportion \( \hat{p} \) is often used as an estimate for the true population proportion \( p \). From the given exercise, the variance of \( \hat{p} \) tells us how much uncertainty or 'wiggle room' we should expect in our estimate.

By comparing variances for different values of \( p \) at a fixed sample size, we can determine which \( p \) value is likely to give us a sample proportion closer to the population proportion. A smaller variance indicates that the sample proportion is more likely to be close to the population proportion, as seen in the exercise where \( \hat{p} \) is closer to the true \( p \) when \( p=0.7 \) than when \( p=0.4 \), due to the smaller resulting variance.

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Most popular questions from this chapter

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 \(p=0.4\) ?

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