91Ó°ÊÓ

Find the sample proportion \(\hat{p}\). The math SAT score is higher than the verbal SAT score for 205 of the 355 students who answered the questions about SAT scores. Find \(\hat{p},\) the proportion for whom the math SAT score is higher.

Give the relevant proportion using correct notation. A recent headline states that "45\% Think Children of Illegal Immigrants Should Be Able to Attend Public School." The report gives the results of a survey of 1000 randomly selected likely voters in the US.

Give the relevant proportion using correct notation. Of all 1,672,395 members of the high school class of 2014 who took the SAT (Scholastic Aptitude Test), 793,986 were minority students.

Data from the StudentSurvey dataset are given. Construct a relative frequency table of the data using the categories given. Give the relative frequencies rounded to three decimal places. Of the 361 students who answered the question about the number of piercings they had in their body, 188 had no piercings, 82 had one or two piercings, and the rest had more than two.

A two-way table is shown for two groups, 1 and \(2,\) and two possible outcomes, A and B. In each case, (a) What proportion of all cases had Outcome \(\mathrm{A}\) ? (b) What proportion of all cases are in Group \(1 ?\) (c) What proportion of cases in Group 1 had Outcome \(\mathrm{B} ?\) (d) What proportion of cases who had Outcome \(\mathrm{A}\) were in Group \(2 ?\) $$\begin{array}{|l|cc|c|}\hline & \text { Outcome A } & \text { Outcome B } & \text { Total } \\ \hline \text { Group 1 } & 40 & 10 & 50 \\ \text { Group 2 } & 30 & 20 & 50 \\\\\hline \text { Total } & 70 & 30 & 100 \\\ \hline\end{array}$$

A recent study shows that antibiotics added to animal feed to accelerate growth can become airborne. Some of these drugs can be toxic if inhaled and may increase the evolution of antibiotic-resistant bacteria. Scientists \(^{9}\) analyzed 20 samples of dust particles from animal farms. Tylosin, an antibiotic used in animal feed that is chemically related to erythromycin, showed up in 16 of the samples. (a) What is the variable in this study? What are the individual cases? (b) Display the results in a frequency table. (c) Make a bar chart of the data. (d) Give a relative frequency table of the data.

Rock-Paper-Scissors, also called Roshambo, is a popular two-player game often used to quickly determine a winner and loser. In the game, each player puts out a fist (rock), a flat hand (paper), or a hand with two fingers extended (scissors). In the game, rock beats scissors which beats paper which beats rock. The question is: Are the three options selected equally often by players? Knowing the relative frequencies with which the options are selected would give a player a significant advantage. A study \(^{10}\) observed 119 people playing Rock-Paper-Scissors. Their choices are shown in Table 2.6 . (a) What is the sample in this case? What is the population? What does the variable measure? (b) Construct a relative frequency table of the results. (c) If we assume that the sample relative frequencies from part (b) are similar for the entire population, which option should you play if you want the odds in your favor? (d) The same study determined that, in repeated plays, a player is more likely to repeat the option just picked than to switch to a different option. If your opponent just played paper, which option should you pick for the next round? $$\begin{array}{lc}\hline \text { Option Selected } & \text { Frequency } \\\\\hline \text { Rock } & 66 \\\\\text { Paper } & 39 \\\\\text { Scissors } & 14 \\\\\hline \text { Total } & 119 \\\\\hline\end{array}$$

Researchers examined all sports-related concussions reported to an emergency room for children ages 5 to 18 in the United States over the course of one year. \({ }^{11}\) Table 2.7 displays the number of concussions in each of the major activity categories. (a) Are these results from a population or a sample? (b) What proportion of concussions came from playing football? (c) What proportion of concussions came from riding bicycles? (d) Can we conclude that, at least in terms of concussions, riding bicycles is more dangerous to children in the US than playing football? Why or why not? $$\begin{array}{l|r}\hline \text { Activity } & \text { Frequency } \\ \hline \text { Bicycles } & 23,405 \\ \text { Football } & 20,293 \\\\\text { Basketball } & 11,507 \\ \text { Playground } & 10,414 \\\\\text { Soccer } & 7,667 \\\\\text { Baseball } & 7,433 \\\\\text { All-Terrain Vehicle } & 5,220 \\\\\text { Hockey } & 4,111 \\\\\text { Skateboarding } & 4,408 \\\\\text { Swimming } & 3,846 \\\\\text { Horseback Riding } & 2,648 \\ \hline \text { Total } & 100,952 \\\\\hline\end{array}$$

A disruption of a gene called \(D Y X C 1\) on chromosome 15 for humans may be related to an increased risk of developing dyslexia. Researchers \({ }^{16}\) studied the gene in 109 people diagnosed with dyslexia and in a control group of 195 others who had no learning disorder. The \(D Y X C 1\) break occurred in 10 of those with dyslexia and in 5 of those in the control group. (a) Is this an experiment or an observational study? What are the variables? (b) How many rows and how many columns will the data table have? Assume rows are the cases and columns are the variables. (There might be an extra column for identification purposes; do not count this column in your total.) (c) Display the results of the study in a two-way table. (d) To see if there appears to be a substantial difference between the group with dyslexia and the control group, compare the proportion of each group who have the break on the \(D Y X C 1\) gene. (e) Does there appear to be an association between this genetic marker and dyslexia for the people in this sample? (We will see in Chapter 4 whether we can generalize this result to the entire population.) (f) If the association appears to be strong, can we assume that the gene disruption causes dyslexia? Why or why not?

In Exercise 1.23, we learned of a study to determine whether just one session of cognitive behavioral therapy can help people with insomnia. In the study, forty people who had been diagnosed with insomnia were randomly divided into two groups of 20 each. People in one group received a one-hour cognitive behavioral therapy session while those in the other group received no treatment. Three months later, 14 of those in the therapy group reported sleep improvements while only 3 people in the other group reported improvements. (a) Create a two-way table of the data. Include totals across and down. (b) How many of the 40 people in the study reported sleep improvement? (c) Of the people receiving the therapy session, what proportion reported sleep improvements? (d) What proportion of people who did not receive therapy reported sleep improvements? (e) If we use \(\hat{p}_{T}\) to denote the proportion from part (c) and use \(\hat{p}_{N}\) to denote the proportion from part (d), calculate the difference in proportion reporting sleep improvements, \(\hat{p}_{T}-\hat{p}_{N}\) between those getting therapy and those not getting therapy.