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AP Statistics 5.4 Biased and Unbiased Point Estimates- MCQs - Exam Style Questions

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Question

If a statistic has a sampling distribution whose mean is different from the true value of the parameter which the statistic is estimating, the statistic is said to be
(A) biased
(B) random
(C) unbiased
(D) efficient
(E) consistent
▶️ Answer/Explanation
Detailed solution

1. Define Unbiased Estimator:
An estimator is considered **unbiased** if the mean of its sampling distribution is equal to the true value of the population parameter it is intended to estimate.

2. Define Biased Estimator:
Conversely, an estimator is **biased** if the mean of its sampling distribution is not equal to the true value of the parameter. The question describes exactly this situation.
✅ Answer: (A)

Question

Researchers will conduct a study of the television-viewing habits of children. They will select a simple random sample of children and record the number of hours of television the children watch per week. The researchers will report the sample mean as a point estimate for the population mean. Which of the following statements is correct for the sample mean as a point estimator?
(A) A sample of size \(25\) will produce more variability of the estimator than a sample of size \(50\).
(B) A sample of size \(25\) will produce less variability of the estimator than a sample of size \(50\).
(C) A sample of size \(25\) will produce a biased estimator, but a sample size of \(50\) will produce an unbiased estimator.
(D) A sample of size \(25\) will produce a more biased estimator than a sample of size \(50\).
(E) A sample of size \(25\) will produce a less biased estimator than a sample of size \(50\).
▶️ Answer/Explanation
Detailed solution

1. Analyze Bias:
The sample mean is an unbiased estimator of the population mean, regardless of the sample size, as long as the sample is random. Therefore, sample size does not affect bias.

2. Analyze Variability:
The variability of the sample mean is measured by its standard deviation, given by the formula \(\sigma_{\bar{x}} = \frac{\sigma}{\sqrt{n}}\). This formula shows that as the sample size (\(n\)) increases, the variability of the sample mean decreases.

Therefore, a smaller sample size (\(n=25\)) will result in a larger standard deviation and thus produce more variability than a larger sample size (\(n=50\)).
✅ Answer: (A)

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