# note that you can load ISCAM.RData to get the iscam* functions
load(url("http://www.rossmanchance.com/iscam3/ISCAM.RData"))Note:
homework (HW) at: http://www.rossmanchance.com/iscam3/instructors.html
Practicee Problems (PP) in the textbook at the end of the investigationsThe Literary Digest
Recall from Investigation 1.13 that the Literary Digest conducted a large sample survey of 2.4 million people and found 57% indicating they would vote for Alf Landon in the upcoming 1936 election.
(a) Use technology to produce a 99.9% confidence interval for the proportion of all voters who were planning to vote for Alf Landon based on this sample data.
iscamonepropztest(phatvalue, samplesize, conf.level=99.9)Representative Samples?
Suppose that you are asked to use the students in your current statistics class as a sample from the population of all students at your school. This is not literally a random sample, but whether or not this sample is representative of the population could depend on the variable of interest. For each of the following variables, discuss whether you think the sample would be representative of the population. Briefly explain your answer. (a) Sex
(b) Time spent sleeping last night
(c) Knowledge of statistics
(d) Political party affiliation
(e) Number of movies seen in past year
(f) Age
Margin-of-Error Properties
The margin-of-error of a confidence interval for a population proportion \(\pi\) using the Wald procedure is \(z^* \sqrt{\pi(1-\pi)/n}\) . Some books recommend a shortcut formula that approximates this margin-of-error for a 95% CI for \(\pi\) quite simply by \(1/\sqrt{n}\).
(a) Explain why this is a reasonable approximation. [Hint: What simplification can you make in the given formula when you are using a 95% confidence level? What about different values of \(\pi\)?]
(b) Show that this approximation is conservative, in that it slightly overestimates the actual margin-of-error.
(c) Reconsider Practice Problem 1.10 and re-answer part (b) – solving for the necessary sample size using this approximation.
(d) Suggest two different ways (that the researcher has direct control over) to reduce the margin-of-error in a study.
Margin-of-Error Properties (cont.)
Consider the margin-of-error of a confidence interval for a population proportion \(\pi\) using the Wald procedure: \(z^*\sqrt{\frac{\hat{p}(1-\hat{p})}{n}}\) . Suppose that you want the margin-of-error to be no larger than some pre-specified error bound, M. That is, we want to satisfy the inequality \(z^*\sqrt{\frac{\hat{p}(1-\hat{p})}{n}} \leq M\).
(a) Rearrange the terms in the inequality to solve for the sample size n necessary to achieve such a margin-of-error, as a function of \(z^*, M\), and \(\hat{p}\).
(b) Is the expression in (a) an increasing function of \(z^*\), a decreasing function of \(z^*\), or neither? Consequently, is it an increasing function of the confidence level, a decreasing function of the confidence level, or neither? Explain why this makes intuitive sense.
(c) Is the expression in (a) an increasing function of M, a decreasing function of M, or neither? Explain why this makes intuitive sense.
(d) Is the expression in (a) an increasing function of \(\hat{p}\), a decreasing function of \(\hat{p}\) , or neither? Use calculus to determine the value of \(\hat{p}\) that maximizes this function. (You can plot the function instead of using calculus here.)
(e) Explain why your answer to (d) suggests that the conservative approach to sample size determination is to use 0.5 as a guess for \(\hat{p}\) .
Leaving Office?
The news website MSNBC.com reports that one week during the Clinton-Lewinsky scandal, readers of the site were invited to vote in an unscientific poll that asked whether President Clinton [Bill, that is] should leave office. The site received over 200,000 votes, of which 73% said “yes.”
(a) Describe the population and parameter of interest here.
(b) Use these sample data to determine a 99% confidence interval for the proportion of adult Americans who felt during the week in question that Clinton should leave office. Also report the margin-of-error.
(c) Do you think that this interval does a reasonable job of estimating the population proportion who felt that Clinton should leave office? Explain. During the same week, an NBC News–Wall Street Journal poll contacted a random sample of 2005 people, with 34% answering that Clinton should leave office.
(d) Re-answer questions (b)–(c) based on these sample data.
(e) Which of these two intervals (the one based on the MSNBC.com poll or the one based on the NBC News- Wall Street Journal poll) is narrower? Explain why that makes sense.
(f) Do these two intervals overlap? Are they similar at all?
(g) Which interval do you think provides a more reasonable estimate of the proportion of adult Americans who felt that Clinton should leave office? Explain.