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Chapter 7: Problem 68

Is the sample proportion a consistent estimator of the population proportion? Explain why or why not.

Short Answer

Expert verified
Yes, the sample proportion is a consistent estimator of the population proportion. This is because, as the sample size increases, the approximation of the population proportion by the sample proportion becomes increasingly accurate, a property embodying the concept of consistency in statistics, according to the law of large numbers.

Step by step solution

01

Understanding the definitions

Firstly, understand the important terms in the problem. A consistent estimator is a statistical property where, as the data set increases to infinity, the estimator tends to the value being estimated. The population proportion is a property that quantifies a characteristic for every individual in a population.
02

Linking the consistency of the estimator to the sample and population

The sample proportion, which is the estimate of the population proportion based on a sample, becomes increasingly accurate at estimating the population proportion as the sample size increases. This is because the sample proportion is a mean statistic, and according to the law of large numbers, the average of the results obtained from a large number of trials should be close to the expected value, and will tend to become closer as more trials are performed.
03

Conclusion

Applying the law of large numbers to our context, it can be inferred that as the sample size grows larger and approaches the size of the entire population, the sample proportion will increasingly approximate to the actual population proportion. Therefore, the sample proportion is a consistent estimator of the population proportion.

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Most popular questions from this chapter

Refer to Exercise 6.92. Otto is trying out for the javelin throw to compete in the Olympics. The lengths of his javelin throws are normally distributed with a mean of 290 feet and a standard deviation of 10 feet. What is the probability that the total length of three of his throws will exceed 885 feet?

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