Introduction to the Practice of Statistics
Sixth Edition
Moore, McCabe
Section 3.3 Homework Answers
3.82b. A statistic will have a large amount of bias whenever it has high variability.
This is false since bias has to do with the sampling method used; does your sampling method systematically favor a certain outcome that is not proportionally the same as the population your are sampling? If it does, then you have bias. Bias is about a procedure. People confuse sample size with bias. A sample of one if done in such a way that this value was not chosen did not favor a particular part of the population is then said not to be bias. The problem with a sample of one is that it has high variability.
For example suppose I wanted to know who was voting for Obama, and my sampling method consistenly favored 18-24 old voters. A sample of 10,000 using this method would have high bias, but very low variability.
(c) The variability of a statistic based on a small sample from a population will be the same as the variability of a large sample from the same population.
This is false. Think about it. If you wanted to estimate the average of a population, which would be more accurate, a sample of 100 or a sample of 1000? Everyone, would say the sample of 1000? But why? Because you recognize that the sample of 1000 should be closer to the true mark (assuming a good sampling method with very little bias). But why would you be closer to the true mark? Afterall, we have already acknowledge that a statistic changes from sample to sample. But you also recognize that the sample of 1000 would not vary as much. Thus the variability of larger sample is less than that of a smaller sample.
3.84
(a) High variability, high bias.
(b) Low variability, low bias.
(c) High variability, low bias.
(d) Low variability, high bias.
3.88 (a) Population: Adults who reside in Ontario Canada. Sample: The 61, 239 adults sampled.
(b) To answer this question you must first answer this other question: Was the sample biased?
Since we do not know the answer to this question we must consider both cases.
Case 1 - No bias or very little bias in sample: Then, the 76% is probably good to the ones digit, and likewise for the 86%. Why? Galllup, needs survey of about a 1000 to get within 3% points. Lets assume that the survey done was about half men, half women. That is roughly 30,000 men and women surveyed. To cut the margin of error in Gallup by half I need a sample of 4000. To cut it in half again, I need sample of 16000, that gets me to (1/2(1/2) of 3% or 0.75%. The actual sample is a little under double of 16000, thus, I am roughly down to a margin of error around 0.6%. So you can see the value of 76% and 86% is right on the actual value.
Case 2 - A sample with considerable bias: Then the two values are not accurate at all. Depending on the bias, we could be way off the actual value. If the sample of 61000 consistenly favored a group that does not feel the same as the entire population (no independence) then the estimated values are no good.