Discuss each of the following topics in class or review the topics on your own. Then write a brief but complete essay in which you summarize the main points. Please include formulas and graphs as appropriate.
1. In this chapter, we have studied confidence intervals. Carefully read the following statements about confidence intervals:
(a) Once the endpoints of the confidence interval are numerically fixed, then the parameter in question (either m or p) does or does not fall inside the “fixed” interval.
(b) A given fixed interval either does or does not contain the parameter m or p, therefore, the probability is 1 or 0 that the parameter is in the interval.
Next, read the following statements. Then discuss all four statements in the context of what we actually mean by a confidence interval.
(c) Nontrivial probability statements can be made only about variables, not constants.
(d) The confidence level c represents the proportion of all (fixed) intervals that would contain the parameter if we repeated the process many, many times.
2. Throughout Chapter 8, we have used the normal distribution, the central limit theorem, or the Student’s t distribution.
(a) Give a brief outline describing how confidence intervals for means use the normal distribution or Student’s t distribution in their basic construction.
(b) Give a brief outline describing how the normal approximation to the binomial distribution is used in the construction of confidence intervals for a proportion p.
(c) Give a brief outline describing how the sample size for a predetermined error tolerance and level of confidence is determined from the normal distribution or the central limit theorem.
3. When the results of a survey or a poll are published, the sample size is usually given, as well as the margin of error. For example, suppose the Honolulu Star Bulletin reported that it surveyed 385 Honolulu residents and 78% said they favor mandatory jail sentences for people convicted of driving under the influence of drugs or alcohol (with margin of error of 3 percentage points in either direction). Usually the confidence level of the interval is not given, but it is standard practice to use the margin of error for a 95% confidence interval when no other confidence level is given.
(a) The paper reported a point estimate of 78%, with margin of error of 3%. Write this information in the form of a confidence interval for p, the population proportion of residents favoring mandatory jail sentences for people convicted of driving under the influence. What is the assumed confidence level?
(b) The margin of error is simply the error due to using a sample instead of the entire population. It does not take into account the bias that might be introduced by the wording of the question, by the truthfulness of the respondents, or by other factors. Suppose the question was asked in this fashion:
“Considering the devastating injuries suffered by innocent victims in auto accidents caused by drunken or drugged drivers, do you favor a mandatory jail sentence for those convicted of driving under the influence of drugs or alcohol?” Do you think the wording of the question would influence the respondents? Do you think the population proportion of those favoring mandatory jail sentences is accurately represented by a confidence interval based on responses to such a question? Explain your answer. 

If the question had been: “Considering existing overcrowding of our prisons, do you favor a mandatory jail sentence for people convicted of driving under the influence of drugs or alcohol?” Do you think the population proportion of those favoring mandatory sentences is accurately represented by a confidence interval based on responses to such a question? Explain.