Discussion:
Q1: Conduct the hypothesis test and provide the test statistic, critical value and/or P-value, and state the conclusion.
Flat Tire and Missed Class - A classic tale involves four carpooling students who missed a test and gave as an excuse a flat tire. On the makeup test, the instructor asked the students to identify the particular tire that went flat. If they really didn't have a flat tire, would they be able to identify the same tire? The author asked 41 other students to identify the tire they would select. The results are listed in the following table (except for one student who selected the spare). Use a 0.05 significance level to test the author's claim that the results fit a uniform distribution. What does the result suggest about the ability of the four students to select the same tire when they really didn't have a flat?
Tire Left front Right front Left rear Right rear
Number selected 11 15 8 6
Benford's Law - According to Benford's Law, a variety of different data sets include numbers with leading (first) digits that follow the distribution shown in the table below., test for goodness-of-fit with Benford's law.
Leading Digit 1 2 3 4 5 6 7 8 9
Benford's Law: 30.1% 17.6% 12.5% 9.7% 7.9% 6.7% 5.8% 5.1% 4.6%
Distribution of leading digits
Q2: Check amounts - In the trial of State of Arizona vs. Wayne James Nelson, the defendant was accused of issuing checks to a vendor that did not really exist. The amounts of the checks are listed below in order by row. When testing for goodness-of-fit with the proportions expected with Benford's law, it is necessary to combine categories because not all expected values are at least 5. Use one category with leading digits of 1, a second category with leading digits of 2, 3, 4, 5, and a third category with leading digits of 6, 7, 8, 9,. Using a 0.01 significance level, is there sufficient evidence to conclude that the leading digits on the checks do not conform to Benford's law?
$1,927.48 $27,902.31 $86,241.90 $72,117.46 $81,321.75 $97,473.96
$93,249.11 $89,658.16 $87,776.89 $92,105.83 $79,949.16 $87,602.93
$96,879.27 $91,806.47 $84,991.67 $90,831.83 $93,766.67 $88,336.72
$94,639.49 $83,709.26 $96,412.21 $88,432.86 $71,552.16
Test the given Claim
Q3: Global Warming Survey - A pew research poll was conducted to investigate opinions about global warming. The respondents who answered yes when asked if there is solid evidence that the earth is getting warmer were then asked to select a cause of global warming. The results for two age brackets are given in the table below. Use a 0.01 significance level to test the claim that the age bracket is independent of the choice for the cause of global warming.
Do respondents from both age brackets appear to agree, or is there a substantial difference?
Human activity Natural patterns Don't know or refused to answer
Under 30 108 41 7
65 and over 121 71 43
Q4: Injuries and Motorcycle Helmet Color - A case-control (ore retrospective) study was conducted to investigate a relationship between the colors of helmets worn by motorcycle drivers and whether they are injured or killed in a crash. Results are given in the table below (based on data from "Motorcycle Rider Conspicuity and Crash Related Injury: Case-Control Study," by Wells, et al., BMJ USA, Vol. 4). Test the claim that injuries are independent of helmet color. Should motorcycle drivers choose helmets with a particular color? If so, which color appears best?
Color of Helmet
Black White Yellow/Orange Red Blue
Controls (not injured) 491 377 31 170 55
Cases (injured or killed) 213 112 8 70 26