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In the 2018 study Closing the STEM Gap, researchers wanted to estimate the percentage of middle school girls who planned to major in a STEM field. a. If a \(95 \%\) confidence level is used, how many people should be included in the survey if the researchers wanted to have a margin of error of \(3 \%\) ? b. How could the researchers adjust their margin of error if they want to decrease the number of study participants?

The Perry Preschool Project was created in the early \(1960 \mathrm{~s}\) by David Weikart in Ypsilanti, Michigan. In this project, 123 African American children were randomly assigned to one of two groups: One group enrolled in the Perry Preschool, and the other group did not. Follow-up studies were done for decades. One research question was whether attendance at preschool had an effect on high school graduation. The table shows whether the students graduated from regular high school or not and includes both boys and girls (Schweinhart et al. 2005 ). Find a \(95 \%\) confidence interval for the difference in proportions, and interpret it. $$ \begin{array}{|lcc|} \hline & \text { Preschool } & \text { No Preschool } \\ \hline \text { Grad HS } & 37 & 29 \\ \hline \text { No Grad HS } & 20 & 35 \\ \hline \end{array} $$

In 2017 , the journal Obesity reported on trends in sugar-sweetened beverage (SSB) consumption. A random sample of youths aged 12 to 19 years old were asked to monitor all food and beverages consumed in a 24 -hour period. The study was done in 2003 and repeated in 2014 . The numbers who consumed a sugary beverage such as soda or fruit juice in a day are shown in the table. (Bleich et al., "Trends in Beverage Consumption among Children and Adults, 2003-2014," Obesity, vol. 26 [2018]: 432-441. doi:10.1002/oby.22056) $$ \begin{array}{|l|l|} \hline \text { Consumed SSB } & \mathbf{2 0 0 3} & \mathbf{2 0 1 4} \\ \hline \text { Yes } & 3416 & 2682 \\ \hline \text { No } & 685 & 1419 \\ \hline \end{array} $$ a. Calculate and compare the percentages of youths in this age group who consumed an SSB during the recording period. b. Check that the conditions for using a two-population confidence interval hold. c. Find the \(95 \%\) confidence interval for the difference in the proportion of youth consuming an SSB in 2003 and 2014. Based on your confidence interval, do you think there has been a change in sugar-sweetened beverage consumption among this age group? Explain.

Statistics student Hector Porath wanted to find out whether gender and the use of turn signals when driving were independent. He made notes when driving in his truck for several weeks. He noted the gender of each person that he observed and whether he or she used the turn signal when turning or changing lanes. (In his state, the law says that you must use your turn signal when changing lanes, as well as when turning.) The data he collected are shown in the table. $$ \begin{array}{|lcc|} \hline & \text { Men } & \text { Women } \\ \hline \text { Turn signal } & 585 & 452 \\ \hline \text { No signal } & 351 & 155 \\ \hline & 936 & 607 \\ \hline \end{array} $$ a. What percentage of men used turn signals, and what percentage of women used them? b. Assuming the conditions are met (although admittedly this was not a random selection), find a \(95 \%\) confidence interval for the difference in percentages. State whether the interval captures 0, and explain whether this provides evidence that the proportions of men and women who use turn signals differ in the population. c. Another student collected similar data with a smaller sample size: $$ \begin{array}{|l|l|l|} \hline & \text { Men } & \text { Women } \\ \hline \text { Turn Signal } & 59 & 45 \\ \hline \text { No Signal } & 35 & 16 \\ \hline & 94 & 61 \\ \hline \end{array} $$ First find the percentage of men and the percentage of women who used turn signals, and then, assuming the conditions are met, find a \(95 \%\) confidence interval for the difference in percentages. State whether the interval captures 0 , and explain whether this provides evidence that the percentage of men who use turn signals differs from the percentage of women who do so. d. Are the conclusions in parts \(\mathrm{b}\) and \(\mathrm{c}\) different? Explain.

A poll on a proposition showed that we are \(95 \%\) confident that the population proportion of voters supporting it is between \(40 \%\) and \(48 \%\). Find the margin of error.

Suppose a political consultant is hired to determine if a school bond is likely to pass in a local election. The consultant randomly samples 250 likely voters and finds that \(52 \%\) of the sample supports passing the bond. Construct a \(95 \%\) confidence interval for the proportion of voters who support the bond. Assume the conditions are met. Based on the confidence interval, should the consultant predict the bond will pass? Why or why not?

Find the sample size required for a margin of error of 3 percentage points, and then find one for a margin of error of \(1.5\) percentage points; for both, use a \(95 \%\) confidence level. Find the ratio of the larger sample size to the smaller sample size. To reduce the margin of error to half, by what do you need to multiply the sample size?