Rossman (& Mine)
Penny Spinning Problem (finding “n”)
Reconsider the penny spinning experiment. Let P denote the proportion of all spins that would land “tails.”
a. Using the sample proportion obtained in class as an initial estimate, determine how many spins would be required to estimate P to within 2% with 95% confidence.
b. Determine how many spins would be required to estimate P to within 2% with 99% confidence.
c. Determine how many spins would be required to estimate P to within 1% with 99.9% confidence.
P-Value Chart
p-value > .10: no evidence against Ho
.05 < p-value .10: some weak evidence against Ho
.01 < p-value .05: fairly strong evidence against Ho
.001 < p-value .01: strong evidence against Ho
p-value .001: very strong evidence against Ho
Lady Tasting Tea Problem
A famous hypothetical example of statistical inference involves a woman who claims that when presented with a cup of tea and milk, she can distinguish more often than not whether it was the tea or the milk that had been poured first. Suppose that you present her with 100 cups of tea and milk which you have prepared (so that you know for each cup whether it was the tea or the milk that was poured first). For each cup you ask her to identify which was poured first.
a) Determine how many correct identifications she must make in order for the sample result to be statistically significant at the .10 level. Approach this question analytically.
b) Repeat (a) for the .05 significance level.
c) Repeat (a) for the .01 significance level.
Dentists’ Surveys Problem
Suppose that a survey of dentists finds that 60% (3 out of every 5) of the dentists sampled recommend a certain toothbrush. Even if we assume that this is a random sample and that the dentists’ opinions are sincere, does this provide strong evidence that more than half of all dentists prefer the toothbrush? Perform a test of significance to address this question for each of the sample sizes listed below. (You may use your TI-83.) In each case report the p-value of the test and indicate whether the sample result is statistically significant at the .10 level, at the .05 level, at the .01 level, and at the .001 level.
Sample Size / One-SidedP-Value / Sig. at
.10 level? / Sig. at
.05 level? / Sig. at
.01 level? / Sig. at
.001 level?
a) n=25
b) n=50
c) n=100
d) n=500
Advertising Strategies Problem
Suppose that the managers of a company want to decide whether more than half of the people who use the company’s product are women, because they are planning to launch a new advertising campaign if they discover that most of their customers are women. They ask you to take a SRS of 200 of their customers and to report only whether the sample proportion of women is significantly greater (at the =.05 significance level) than one-half.
a) Suppose that 111 of the 200 customers sampled are women. Determine (using your calculator) the p-value of the appropriate significance test. Report the test statistic and p-value; also indicate whether or not the sample proportion is statistically significantly (at the .05 level) greater than one-half.
b) Repeat (a), but supposing that 112 of the customers sampled are women.
c) Repeat (a), but supposing that 124 of the customers sampled are women.
d) Are the sample results more similar in (a) and (b), or in (b) and (c)?
e) Would your report to the company be the same in (a) and (b), or in (b) and (c)?
The moral here is that it is unwise to treat standard significance levels as sacred. It is much more informative to consider the p-value of the test and to base one’s decision on it. There is no sharp border between “significant” and “insignificant,” only increasingly strong evidence as the p-value decreases. If you were responsible for analyzing the data and reporting to the company’s managers, you would provide them with the best information by giving them the sample proportion and the
p-value of the test (along with an explanation of what the p-value means). On the other hand, if you were one of the managers, you would serve your company best by asking for that information instead of just a statement of significance.
Lunch Pass Check Survey Problem
I. The Problem
Suppose you want to take a SRS of RHS students regarding the issue of whether they favor the checking of off-campus lunch passes. Your question: “Are you in favor of checking lunch passes?”
II. Survey Method (Comment on each method.)
a) Stand next to the gate with a clipboard during lunch.
b) Go into classrooms and have a show of hands.
c) Randomly generate Student-ID #’s and then call or contact students.
III. The Count
How many students must you survey to build a 90% CI and have a margin of error within 6 percentage points?
IV. The Tests
Run two tests of significance (Hypothesis and CI “by hand”) to determine if less than half of all RHS students favor the new policy. State necessary assumptions. (I can think of three.) Suppose 81 out of 188 students are in favor of the checking policy. Do a complete write-up stating your conclusions using clear and concise language.