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Chapter 14: Problem 32
List the characteristics of a multinomial experiment.
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According to the genetic model for the relationship between sex and color blindness, the four categories, male and normal, female and normal, male and color blind, female and color blind, should have probabilities given by \(p / 2,\left(p^{2} / 2\right)+p q, q / 2,\) and \(q^{2} / 2,\) respectively, where \(q=1-p . A\) sample of 2000 people revealed \(880,1032,80,\) and 8 in the respective categories. Do these data agree with the model? Use \(\alpha=.05 .\) (Use maximum likelihood to estimate \(p\).)
Historically, the proportions of all Caucasians in the United States with blood phenotypes A, B, AB, and 0 are \(.41, .10, .04,\) and \(.45,\) respectively. To determine whether current population proportions still match these historical values, a random sample of 200 American Caucasians were selected, and their blood phenotypes were recorded. The observed numbers with each phenotype are given in the following table. $$\begin{array}{cccc}\mathbf{A} & \mathbf{B} & \mathbf{A B} & \mathbf{0} \\\\\hline 89 & 18 & 12 & 81 \\\\\hline\end{array}$$ a. Is there sufficient evidence, at the .05 level of significance, to claim that current proportions differ from the historic values? b. Use the applet Chi-Square Probability and Quantiles to find the \(p\) -value associated with the test in part (a).
The Mendelian theory states that the number of a type of peas that fall into the classifications round and yellow, wrinkled and yellow, round and green, and wrinkled and green should be in the ratio \(9: 3: 3: 1 .\) Suppose that 100 such peas revealed \(56,19,17,\) and 8 in the respective categories. Are these data consistent with the model? Use \(\alpha=.05 .\) (The expression 9:3:3:1 means that 9/16 of the peas should be round and yellow, \(3 / 16\) should be wrinkled and yellow, etc.)
A genetic model states that the proportions of offspring in three classes should be \(p^{2}, 2 p(1-p)\) and \((1-p)^{2}\) for a parameter \(p, 0 \leq p \leq 1 .\) An experiment yielded frequencies of \(30,40,\) and 30 for the respective classes. a. Does the model fit the data? (Use maximum likelihood to estimate \(p\).) b. Suppose that the hypothesis states that the model holds with \(p=.5 .\) Do the data contradict this hypothesis?
Do you hate Mondays? Researchers in Germany have provided another reason for you: They concluded that the risk of heart attack on a Monday for a working person may be as much as \(50 \%\) greater than on any other day. The researchers kept track of heart attacks and coronary arrests over a period of 5 years among 330,000 people who lived near Augsberg, Germany. In an attempt to verify the researcher's claim, 200 working people who had recently had heart attacks were surveyed. The day on which their heart attacks occurred appear in the following table. $$\begin{array}{ccccccc}\hline \text { Sunday } & \text { Monday } & \text { Tuesday } & \text { Wednesday } & \text { Thursday } & \text { Friday } & \text { Saturday } \\\\\hline 24 & 36 & 27 & 26 & 32 & 26 & 29 \\\\\hline\end{array}$$ Do these data present sufficient evidence to indicate that there is a difference in the percentages of heart attacks that occur on different days of the week? Test using \(\alpha=.05\).
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