Chapter 12: Chi-Square Procedures

Freshmen Politics. The Higher Education Research Institute of the University of California, Los Angeles, publishes information on the characteristics of incoming college freshmen in The American Freshman. Incoming freshmen characterized their political views as liberal, as moderate, and as conservative, For this year, a random sample of 500 incoming college freshmen yielded the following frequency distribution for political views. Freshmen Politics. The Higher Education Research Institute of the University of California, Los Angeles, publishes information on the characteristics of incoming college freshmen in The American Freshman. Incoming freshmen characterized their political views as liberal, as moderate, and as conservative, For this year, a random sample of 500 incoming college freshmen yielded the following frequency distribution for political views.

a. Identify the population and variable under consideration here.

b. At the 55 significance level, do the data provide sufficient evidence to conclude that this year's distribution of political views for incoming college freshmen has changed from the 2000 distribution?

c. Repeat part (b), using a significance level of.

Road Rage. The report Controlling Road Rage: A Literature Review and Pilot Study was prepared for the AAA Foundation for Traffic Safety by D. Rathbone and. Huckabee. The authors discussed the results of a literature review and pilot study on how to prevent aggressive driving and road rage. Road rage is defined as an incident in which an angry or impatient motorist or passenger intentionally injures or kills another motorist, passenger, or pedestrian, or attempts or threatens to injure or kill another motorist, passenger, or pedestrian." One aspect of the study was to investigate road rage as a function of the day of the week. The following table provides a frequency distribution for the days on which 69 road-rage incidents occurred.

At the significance level, do the data provide sufficient evidence to conclude that road-rage incidents are more likely to occur on some days than on others?

In each of Exercises $12.24-12.33$, apply the chi-square goodness-of-fit test, using either the critical-value approach or the$P-$value approach, to perform the required hypothesis test.
Observing that the proportion of blue M&Ms in his bowl of candy appeared to be less than that of the other colors, R. Fricker, Jr., decided to compare the color distribution in randomly chosen bags of M&Ms to the theoretical distribution reported by M&M/MARS consumer affairs. Fricker published his findings in "The Mysterious Case of the Blue M&Ms" (Chance, Vol. 9(4), pp. 19-22). The following table gives the theoretical distribution.
For his study, Fricker bought three bags of M&Ms from local stores and counted the number of each color. The average number of each color in the three bags was distributed as follows.

Do the data provide sufficient evidence to conclude that the color distribution of M&Ms differs from that reported by M&M/MARS consumer affairs? Use $\alpha =0.05$.

In each of Exercises 12.24-12.33, apply the chi-square goodness-of-fit test, using either the critical-value approach or the$P-$value approach, to perform the required hypothesis test.
An American roulette wheel contains $18$red numbers, $18$black numbers, and $2$green numbers. The following table shows the frequency with which the ball landed on each color in $200$trials.

At the $5\%$significance level, do the data suggest that the wheel is out of balance?

Loaded Die? A gambler thinks a die may be loaded, that is, that the six numbers are not equally likely. To test his suspicion, he rolled the die 150 times and obtained the data shown in the following table.

Bottled Water. A project exploring the bottled-water phenomenon and preference of water types was conducted by researchers M. Lunsford and A. Fink in the article "Water Taste Test Data" (Journal of Statistics Education, Vol. 18, No. 1). One hundred nine subjects participated in double-blind taste tests of three different bottled water brands (Fiji, Aquafina, and Sam's Choice) and tap water. Twelve people preferred the tap water, 27 Aquafino 44 Fiji, and 26 Sam's Choice. At the 5% significance level, do the data provide sufficient evidence to conclude that the four different water types are not equally likely in preference?

Consider two ${\chi }^2$curves with degrees of freedom $12$and $20$respectively. which one more closely resembles a normal curve? Explain your answer.

The world series in baseball is won by the first team to win four games (ignoring the \(1903\) and \(1919-1921\) world series, when it was best of nine). Thus it takes at least four games and no more than seven games to establish a winner. If two teams are evenly matched the probabilities of the series lasting \(4, 5, 6, or 7\) games are as given in the second column of the following table. From the document World Series History on the Baseball Almanac website, as of November \(2013\), the actual numbers of times that the series lasted \(4, 5, 6, or 7\) games are as shown in the third column of the table.

a. At the \(1%\) significance level, do the data provide sufficient evidence to conclude that the distribution of genders in two-children families differs from the distribution predicted by the model described?

b. In view of your result from part (a) what conclusion can you draw?

Variegated Plants. Arabidopsis is a genus of flowering plants related to cabbage. A variegated mutant of the Arabidopsis has yellow streaks or marks. E. Miura et al. studied the origin of this variegated mutant in the article "The Balance between Protein Synthesis and Degradation in Chloroplasts Determines Leaf Variegation in Arabidopsis yellow variegated Mutants" (The Plant Cell, Vol. 19, No. 4, pp. 1313-1328). In a second-generation cross of variegated plants, 216 were variegated and 84 were normal. Genetics predicts that 75% of crossed variegated plants would be variegated and 25% would be normal. At the 10% significance level, do the data provide sufficient evidence to conclude that the second generation of crossed variegated plants does not follow the genetic predictions?

Girls and Boys. One probability model for child gender is that a boy or a girl is equally likely to be born. If that model is correct, then, for a two-child family, the probabilities are 0.25,0.50, and 0.25 for two girls, one girl and one boy, and two boys, respectively. W. Stansfield and M. Carlton examined data collected in the National Health Interview Study on two-child families in the article "The Most Widely Published Gender Problem in Human Genetics" (Human Biology, Vol. 81, No. 1, pp. 3-11). Of 42,888 families with exactly two children, 9,523 had two girls, 22,031 had one girl and one boy, and 11,334 had two boys.

a) At the 1% significance level, do the data provide sufficient evidence to conclude that the distribution of genders in two-children families differs from the distribution predicted by the model described?

b) In view of your result from part (a), what conclusion can you draw?