AP Statistics Quiz: Biased And Unbiased Point Estimates

What this quiz covers

This quiz focuses on Biased And Unbiased Point Estimates, giving you a quick way to practice the rules, question types, and explanations that matter most for AP Statistics.

How to use this quiz

Try each quiz question before looking at the correct answer. Use the explanations to review missed ideas, then come back to similar questions until the pattern feels familiar.

Question 1

1. The estimate is biased because self-reported data increase variability, so $\bar{x}$ will tend to be too high.
2. The estimate is unbiased because for an SRS, $E(\bar{x})=\mu$. (correct answer)
3. The estimate is biased because $\bar{x}$ is only unbiased when the population is normally distributed.
4. The estimate is biased because samples of size 60 are not large enough to avoid bias.
5. The estimate is unbiased only if the sample mean equals the population mean exactly.

Explanation: This question tests understanding of unbiased estimators for population means. When we have a simple random sample (SRS), the sample mean $\bar{x}$ is an unbiased estimator of the population mean $\mu$, meaning that if we repeated the sampling process many times, the average of all those sample means would equal $\mu$. The fact that data is self-reported doesn't create bias in the estimator itself - it might affect the accuracy of individual values, but the sample mean from an SRS still targets the true population mean. Choice A incorrectly confuses variability with bias; increased variability affects precision but not whether the estimator is unbiased. Remember: bias is about whether an estimator systematically over- or underestimates the parameter in the long run, while variability is about how spread out the estimates are.

Question 2

1. The estimate is unbiased because $\bar{x}$ is a sample mean, and sample means are always unbiased.
2. The estimate is biased because the net tends to miss lighter fish, so the sample mean will tend to be larger than $\mu$. (correct answer)
3. The estimate is unbiased because catching 70 fish is a large sample, which eliminates bias.
4. The estimate is biased because the sample mean will not equal the population mean exactly in a single sample.
5. The estimate is unbiased only if the fish weights are normally distributed.

Explanation: This question illustrates selection bias through a flawed sampling mechanism. The net's design systematically excludes lighter fish that slip through the holes, meaning the sample only includes heavier fish. This creates an upward bias where $\bar{x}$ will consistently overestimate the true population mean weight $\mu$. No matter how many times the biologist repeats this sampling method, the average of all sample means will be larger than $\mu$ because lighter fish are systematically excluded. Choice A incorrectly assumes all sample means are unbiased, ignoring how the sample was obtained. Choice C wrongly claims large samples eliminate bias, but catching 700 or 7,000 fish with the same net would still exclude the lighter ones. The bias comes from the systematic exclusion of part of the population, not from random sampling variability.

Question 3

1. The estimate is biased because $\hat p$ will vary from sample to sample.
2. The estimate is unbiased because $\hat p$ is an unbiased estimator of $p$ for a simple random sample. (correct answer)
3. The estimate is biased because using a sample instead of a census always underestimates $p$.
4. The estimate is unbiased only if the sample size is much larger than 200; otherwise it is biased.
5. The estimate is biased because increasing the sample size would change the expected value of $\hat p$.

Explanation: This question tests understanding of biased versus unbiased point estimates. An estimator is unbiased if its expected value (long-run average) equals the population parameter it estimates. The sample proportion $\hat{p}$ from a simple random sample is an unbiased estimator of the population proportion $p$, meaning that if we took many random samples and computed $\hat{p}$ for each, the average of all these $\hat{p}$ values would equal $p$. Choice A incorrectly confuses variability with bias—having sample-to-sample variation doesn't make an estimator biased. Remember: bias is about whether the estimator is systematically off-target on average, while variability is about how spread out the estimates are.

Question 4

1. The estimate is unbiased because any sample mean $\bar x$ is an unbiased estimator of $\mu$ regardless of how the sample is chosen.
2. The estimate is biased because surveying early arrivals is not a random sample and could systematically misrepresent commute times. (correct answer)
3. The estimate is unbiased because the sample size is 50, which is large enough to eliminate bias.
4. The estimate is biased because $\bar x$ has sampling variability, so it cannot be unbiased.
5. The estimate is unbiased because bias depends only on the population standard deviation, not on the sampling method.

Explanation: This question tests recognizing bias from non-random sampling methods. The HR manager surveys only the first 50 employees to arrive, which is not a random sample of all employees. Early arrivals might have systematically different commute times (perhaps shorter commutes make it easier to arrive early), so this sampling method could produce a biased estimate. Choice B correctly identifies this bias. Choice A is wrong because being an unbiased estimator requires random sampling from the target population. Remember: the formula for $\bar{x}$ doesn't create bias, but the sampling method can—if your sample systematically excludes or overrepresents certain groups, your estimate will be biased regardless of what statistic you calculate.

Question 5

1. The estimate is unbiased because adding a constant does not affect the center of the sampling distribution.
2. The estimate is biased high because its expected value is $\mu+5$. (correct answer)
3. The estimate is biased low because its expected value is $\mu-5$.
4. The estimate is unbiased for large samples, but biased for small samples like $n=40$.
5. The estimate is biased because $\bar{x}$ has sampling variability.

Explanation: This question evaluates understanding of biased and unbiased estimators, particularly how modifying the sample mean affects bias in estimating mu. Adding 5 to ar{x} shifts the entire sampling distribution by 5, so the expected value becomes mu + 5, making it biased high; over many SRSs, the long-run average of (ar{x} + 5) would be mu + 5, not mu. This intentional adjustment introduces systematic error, even though ar{x} itself is unbiased. Choice E is a distractor that confuses bias with variability, as the variability of ar{x} exists but doesn't cause bias—the addition does. Bias vs. variability mini-lesson: Bias is a fixed offset in the center of the sampling distribution from the true parameter, independent of sample size, while variability is the dispersion that shrinks as n increases, but here the bias persists regardless of n=40. Hence, ar{x} + 5 is biased high.

Question 6

1. The estimate is unbiased because the original selection was an SRS of 80 adults.
2. The estimate may be biased because nonresponse can make the measured group systematically different from the selected sample. (correct answer)
3. The estimate is unbiased because nonresponse only increases variability, not bias.
4. The estimate is biased, but the bias disappears if the measured sample size is still at least 30.
5. The estimate is unbiased only if the blood pressure distribution is normal.

Explanation: This question evaluates bias from nonresponse in an SRS when estimating mu. With 25% nonresponse, the measured group's mean may differ systematically from the full sample, potentially biasing the estimate if non-respondents have different blood pressures; long-run repetitions could center away from mu due to this self-selection. Nonresponse bias affects the sampling distribution's center. Choice C is a distractor stating nonresponse only increases variability, but it can also introduce bias. Bias vs. variability mini-lesson: Bias systematically shifts the center and isn't fixed by remaining sample size, while variability grows with nonresponse but is separate—here, bias may arise from who responds. Therefore, the estimate may be biased.

Question 7

1. The estimate is unbiased because the 10 graded quizzes were randomly selected.
2. The estimate is unbiased because a mean is always an unbiased estimator of a population mean.
3. The estimate may be biased because excluding absent students can systematically change the average compared with all students. (correct answer)
4. The estimate is biased only if the sample size is less than 30; with 10 it is biased, but with 30 it would be unbiased.
5. The estimate is biased because the sample mean varies from sample to sample.

Explanation: This question tests recognizing potential bias from excluding absent students when estimating mu via selected quizzes. Excluding absentees may systematically alter the mean if their scores differ (e.g., lower), so the expected value might not equal mu; over many selections, the long-run average could deviate due to this undercoverage. The method doesn't ensure the sampling distribution centers at mu. Distractor choice E mistakes variability for bias, as variation exists but doesn't cause the systematic shift. Mini-lesson: Bias displaces the center from the parameter due to flawed sampling and persists regardless of size (here 10), whereas variability is spread that reduces with larger n but can't eliminate bias. Thus, the estimate may be biased.

Question 8

1. The estimate is unbiased because multiplying by 2 reduces sampling variability.
2. The estimate is biased high because its expected value is $2\mu$ rather than $\mu$. (correct answer)
3. The estimate is unbiased because $\bar{x}$ is unbiased and scaling does not change bias.
4. The estimate is unbiased only when $\mu=0$.
5. The estimate is biased, but the bias disappears for $n=100$.

Explanation: This question tests identifying bias in modified estimators, like scaling the sample mean 2ar{x} for mu. Multiplying ar{x} by 2 scales the expected value to 2mu, creating a high bias; over many SRSs, the long-run average would be 2mu, not mu, due to this arbitrary adjustment. The sampling distribution's center is shifted proportionally. Distractor choice C incorrectly states scaling doesn't affect bias, but it does when the factor isn't 1. Mini-lesson: Bias is a systematic deviation in the estimator's expected value from the parameter, unaffected by sample size n, whereas variability measures spread and reduces with larger n—here, n=100 lowers variability but the bias remains 2mu - mu. Thus, 2ar{x} is biased high.

Question 9

1. The estimate is unbiased because $\hat{p}$ is always an unbiased estimator of $p$ regardless of how the sample is selected.
2. The estimate is biased because using a convenience sample can systematically overestimate or underestimate $p$. (correct answer)
3. The estimate is unbiased because the sample size is large, which eliminates bias.
4. The estimate is biased because $\hat{p}$ varies from sample to sample, so it cannot be unbiased.
5. The estimate is unbiased only if the sample proportion equals the true proportion in the population.

Explanation: This question addresses bias from non-random sampling methods. A convenience sample, where inspectors only visit homes on two nearby streets, is not representative of all households in the city. Homes on these particular streets might have systematically different smoke detector rates than the city as a whole - perhaps they're in a newer neighborhood with stricter building codes, or an older area where detectors are less common. This sampling method will cause the sample proportion $\hat{p}$ to systematically over- or underestimate the true population proportion $p$ in the long run. Choice A is wrong because the sampling method matters greatly for bias. Choice C incorrectly claims that large samples eliminate bias, but no sample size can fix the fundamental problem of non-representative sampling. Bias is about the sampling method's systematic tendency, not sample-to-sample variability.

Question 10

1. The estimate is unbiased because $\bar x$ is an unbiased estimator of $\mu$ when the sample is selected at random. (correct answer)
2. The estimate is biased because the sample size is only 30, which is too small to avoid bias.
3. The estimate is biased because $\bar x$ will not equal $\mu$ exactly for most samples.
4. The estimate is unbiased only if the population distribution of weights is approximately normal.
5. The estimate is biased because random sampling reduces variability but not bias.

Explanation: This question asks about bias in estimating a population mean. The sample mean $\bar{x}$ from a simple random sample is an unbiased estimator of the population mean $\mu$, regardless of sample size or population distribution shape. This means that over many random samples, the average of all sample means equals the population mean. Choice C incorrectly suggests that because $\bar{x}$ won't exactly equal $\mu$ for most samples, the estimator is biased—but bias is about long-run average behavior, not individual sample results. The key distinction: an unbiased estimator can still produce estimates that differ from the true parameter value; it just doesn't systematically over- or underestimate on average.

Question 11

1. The estimate is unbiased because $\hat p$ is unbiased for $p$ whenever the sample size is at least 30.
2. The estimate is biased because the sampling method (near-shore, one afternoon) may not represent all fish in the lake. (correct answer)
3. The estimate is unbiased because the biologist tested a fairly large number of fish, so bias is eliminated.
4. The estimate is biased because $\hat p$ has sampling variability.
5. The estimate is unbiased only if the biologist repeats the same procedure on multiple afternoons and takes the largest $\hat p$.

Explanation: This question tests recognizing sampling bias from a non-representative collection method. The biologist catches fish using a net near the shoreline during one afternoon, which is not a random sample of all fish in the lake. Fish near shore might differ from those in deeper water, and afternoon behavior might differ from other times. This systematic sampling method could produce a biased estimate of the infection rate. Choice B correctly identifies this bias. Choice A incorrectly suggests sample size alone determines bias—but even large samples can be biased if not randomly selected. Remember: bias comes from systematic problems in how you collect data, not from random sampling variability.

Question 12

1. The estimate is unbiased because the sample mean $\bar x$ is always an unbiased estimator of $\mu$ regardless of how the bulbs were selected.
2. The estimate is biased because selecting only bulbs that passed an initial screen can systematically increase the average lifetime compared with all bulbs produced. (correct answer)
3. The estimate is unbiased because testing 25 bulbs is enough to make the estimate accurate and unbiased.
4. The estimate is biased because a larger sample would reduce the standard deviation of $\bar x$.
5. The estimate is unbiased only if the engineer uses the sample median instead of $\bar x$.

Explanation: This question examines bias from selecting a non-representative subset. The engineer tests only bulbs that passed an initial quality screen, not a random sample of all bulbs produced. Bulbs that pass quality screening likely have longer lifetimes than the general production, so this sampling method will systematically overestimate the population mean lifetime. Choice B correctly identifies this selection bias. Choice A is wrong because $\bar{x}$ is only unbiased when computed from a random sample of the target population. The lesson: when your sample systematically excludes certain types of observations (here, lower-quality bulbs), your estimate will be biased regardless of your calculation method.

Question 13

1. The estimate is unbiased because a randomly selected sample mean $\bar x$ has expected value $\mu$. (correct answer)
2. The estimate is biased because only 40 tests were chosen, and small samples always produce biased estimates.
3. The estimate is biased because $\bar x$ will almost never exactly equal $\mu$.
4. The estimate is unbiased only if the teacher uses the median instead of the mean.
5. The estimate is biased because random sampling reduces bias only when the population is normal.

Explanation: This question asks about bias when using a random sample to estimate a population mean. Since the teacher randomly selects 40 test papers from all completed tests, the sample mean $\bar{x}$ is an unbiased estimator of the population mean $\mu$. Choice A correctly states this. Choice B incorrectly claims small samples produce biased estimates—sample size affects precision (variability) but not bias. Choice C confuses the fact that individual estimates differ from the parameter with bias, which is about long-run average behavior. The key principle: random sampling from the target population produces unbiased estimators, regardless of sample size or whether individual estimates equal the true parameter.

Question 14

1. The estimate is biased because blood pressure measurements are naturally variable, so $\bar x$ cannot be unbiased.
2. The estimate is unbiased because a random sample mean $\bar x$ has expected value $\mu$ for the defined population of adult visits this month. (correct answer)
3. The estimate is biased because the sample size 60 is not large enough to guarantee unbiasedness.
4. The estimate is unbiased only if the distribution of blood pressure is normal.
5. The estimate is biased because using a random number generator tends to select extreme observations more often.

Explanation: This question asks about bias when using proper random sampling. The clinic uses a random number generator to select patient records, which gives each adult visit an equal chance of selection—this is simple random sampling. For such a sample, the sample mean $\bar{x}$ is an unbiased estimator of the population mean $\mu$. Choice B correctly identifies this. Choice A incorrectly suggests that natural variability in measurements creates bias—variability and bias are different concepts. Choice D wrongly claims that unbiasedness requires normality—the sample mean is unbiased regardless of the population distribution shape. Remember: unbiased means the estimator is on-target on average across many samples, not that it has low variability.

Question 15

1. The estimate is unbiased because a large number of responses will make it accurate.
2. The estimate is unbiased because it is a proportion, and proportions are always unbiased.
3. The estimate may be biased because voluntary response samples can systematically overrepresent strong opinions. (correct answer)
4. The estimate is unbiased because emailing all households gives every household an equal chance to be included.
5. The estimate is biased only if the sample size is small; otherwise, it is unbiased.

Explanation: This question probes the skill of recognizing bias in point estimates, especially from non-random sampling like voluntary response for estimating proportion p. Voluntary response samples can introduce bias by overrepresenting those with strong opinions, so the expected value of the respondent proportion may not equal p; long-run repetitions might systematically overestimate or underestimate due to self-selection. Unlike an SRS, this method doesn't ensure the sampling distribution centers at p. Choice A is a distractor assuming large responses eliminate bias, but bias stems from the sampling method, not size. Bias vs. variability mini-lesson: Bias shifts the center away from the parameter due to flawed methods and persists even with large samples, while variability decreases with more data but can't correct bias—here, voluntary response likely biases the estimate. Therefore, the estimate may be biased.

Question 16

1. The estimate is unbiased because $E(\hat{p})=p$ for an SRS. (correct answer)
2. The estimate is biased because $\hat{p}$ is random and changes with each sample.
3. The estimate is biased unless the sample size is at least 500.
4. The estimate is unbiased only when the sample proportion is close to 0.5.
5. The estimate is biased, but increasing $n$ eliminates the bias.

Explanation: This question tests the concept of biased and unbiased point estimates, specifically for the sample proportion hat{p} estimating the population proportion p. For an SRS, hat{p} is unbiased because the expected value E(hat{p}) equals p, so in the long run, across many repeated samples, the average of the hat{p} values would equal the true p. The sampling distribution of hat{p} is centered at p, confirming no systematic over- or underestimation. A frequent distractor is choice B, which mistakes the randomness and variability of hat{p} for bias; variability is the spread of the distribution, not a shift in its center. Mini-lesson: Bias occurs when the center of the sampling distribution deviates from the parameter, whereas variability measures how much estimates fluctuate around that center and can be reduced by increasing sample size n, but bias remains unchanged by n. Therefore, hat{p} is unbiased in this scenario.

Question 17

1. The estimate is unbiased because trimming reduces variability without affecting the center.
2. The estimate is biased high because removing the smallest values tends to increase the expected value of the estimator. (correct answer)
3. The estimate is unbiased because the sample was random.
4. The estimate is biased low because removing values decreases the sample size.
5. The estimate is unbiased for large samples, and $n=50$ is large enough.

Explanation: This question examines bias in point estimates, focusing on a trimmed mean that discards only the smallest values when estimating mu. Removing the 5 smallest from 50 shifts the mean higher, so the expected value exceeds mu, introducing high bias; long-run averages over repeated SRSs would be systematically inflated. This asymmetric trimming alters the center of the sampling distribution. Choice A is a distractor claiming trimming reduces variability without bias, but while it may lower variability, it biases unless symmetrically applied to a symmetric distribution. Bias vs. variability mini-lesson: Bias offsets the center and isn't mitigated by n=50, while variability shrinks with n but here the bias persists from the method. Hence, the trimmed mean is biased high.

Question 18

1. The estimate is unbiased because rounding introduces random error with mean 0, so the expected value remains $\mu$. (correct answer)
2. The estimate is biased high because rounding always increases measurements.
3. The estimate is biased low because rounding always decreases measurements.
4. The estimate is biased because $n=30$ is too small; a larger sample would remove the bias.
5. The estimate is biased because the sample mean has variability from sample to sample.

Explanation: This question assesses whether rounding measurements introduces bias in the sample mean for estimating mu. Rounding to the nearest 2 inches adds errors between -1 and +1 inch; if fractional parts are uniformly distributed, the expected error is 0, so E(rounded ar{x}) = mu, making it unbiased—long-run averages would still hit mu. The sampling distribution remains centered at mu despite added noise. Distractor choice E confuses bias with variability, as sample-to-sample changes don't shift the center. Mini-lesson: Bias occurs if the center deviates systematically, but here it doesn't; variability increases slightly from rounding but decreases with n=30, distinguishing spread from center shift. Thus, the estimate is unbiased.

Question 19

1. The estimate is biased because $\bar{x}$ will vary from sample to sample.
2. The estimate is unbiased because the mean of $\bar{x}$ over many SRSs equals $\mu$. (correct answer)
3. The estimate is biased because using a larger sample size would change the center of the sampling distribution.
4. The estimate is unbiased only if the population distribution of sleep hours is approximately normal.
5. The estimate is biased, but the bias goes to 0 when $n=60$ is large.

Explanation: This question assesses the skill of identifying biased and unbiased point estimates in AP Statistics, focusing on the sample mean as an estimator for the population mean. An unbiased estimator has a sampling distribution whose mean equals the true population parameter, meaning that over many repeated simple random samples (SRSs), the average of the sample means ar{x} would equal mu. Here, since the sample mean ar{x} from an SRS is unbiased for mu, the long-run average of many ar{x} values would center at mu, regardless of sample-to-sample variability. A common distractor is choice A, which confuses bias with variability; variability means ar{x} changes across samples, but that doesn't make it biased—the center still hits mu. In a mini-lesson on bias versus variability: bias refers to a systematic shift in the center of the sampling distribution away from the parameter, while variability describes the spread around that center, which decreases with larger sample sizes but doesn't affect bias. Thus, ar{x} is unbiased here.

Question 20

1. The estimate is unbiased because random sampling prevents bias in measurement.
2. The estimate is biased high because all measurements are shifted upward by 0.2 kg. (correct answer)
3. The estimate is biased low because $n=25$ is too small.
4. The estimate is unbiased, but it will have high variability because $n=25$ is small.
5. The estimate is unbiased because the bias disappears as sample size increases.

Explanation: This question addresses biased and unbiased point estimates, examining how measurement error impacts the sample mean's bias for mu. The scale's consistent 0.2 kg overreading shifts every weight upward, so the sample mean is biased high with expected value mu + 0.2; in the long run, repeated samples would average to mu + 0.2, not mu. This is a systematic measurement bias affecting the center of the sampling distribution. Distractor choice D mixes up bias and variability, noting high variability from small n=25, but variability doesn't cause bias—the overreading does. Mini-lesson on bias vs. variability: Bias systematically displaces the sampling distribution's mean from the parameter and isn't reduced by larger n, whereas variability is the spread that does decrease with n, but here the bias remains fixed at +0.2 kg. Thus, the estimate is biased high.