Exam 8: Analyzing Probabilities and Expected Values

Suppose that you have a $1 / 9$ probability of winning a game of solitaire, independently from game to game. -If you play two games, what is the probability that you lose both? State which probability rule(s) you use for this calculation.

When betting on a color in roulette, you have an 18/38 probability of winning on any one spin. Suppose that you bet on a color for five spins and count the number of times that you win. -Determine the binomial probability that you win at least half of your spins if you spin 30 times.

Suppose that 70% of the customers at a particular fast food restaurant order a hamburger, 50% order French fries, and 45% order both. Let H denote the event that a randomly selected customer orders a hamburger, and let F denote the event that he or she orders French fries. -Determine the conditional probability that a randomly selected customer orders a hamburger given that he or she orders French fries.

Suppose you encounter two traffic lights on your commute to school. You estimate that the probability is .5 that the first light will be red when you get to it, .4 that the second light will be red, and .6 that at least one of the lights will be red. -Consider the random variable, which is the number of red lights that you encounter on this commute. Determine its probability distribution by listing its possible values and their probabilities.

A group of students asked participants from their school to see how many sit-ups they could perform in one minute. Subjects were then randomly assigned to drink either Sobe or Rockstar, two energy drinks. Rockstar is marketed primarily as an energy drink, whereas Sobe is advertised more for taste. Exactly 30 minutes later, the number of sit-ups in one minute was again recorded. The students conjectured that the increase in the number of sit-ups performed would be higher for students drinking Rockstar than for Sobe. -There are 19 males but only 12 females in this study. Explain why gender is not necessarily a confounding variable in this study.

A group of students asked participants from their school to see how many sit-ups they could perform in one minute. Subjects were then randomly assigned to drink either Sobe or Rockstar, two energy drinks. Rockstar is marketed primarily as an energy drink, whereas Sobe is advertised more for taste. Exactly 30 minutes later, the number of sit-ups in one minute was again recorded. The students conjectured that the increase in the number of sit-ups performed would be higher for students drinking Rockstar than for Sobe. -If you had instead conducted a two-sided $t$ -test to compare the two groups, specify the value of the $p$ -value you would have obtained.

Data were collected on the weights for the entire roster of the 2005 Cal Poly football team. Individuals were classified as playing offense, defense, or special teams. Suppose you want to consider these data as representative of all collegiate football players and want to compare the average weight of the players in these three groups. -Summarize the conclusion that you would draw from this study, with regard to the question of whether weights tend to differ across these categorizations.

A group of students asked participants from their school to see how many sit-ups they could perform in one minute. Subjects were then randomly assigned to drink either Sobe or Rockstar, two energy drinks. Rockstar is marketed primarily as an energy drink, whereas Sobe is advertised more for taste. Exactly 30 minutes later, the number of sit-ups in one minute was again recorded. The students conjectured that the increase in the number of sit-ups performed would be higher for students drinking Rockstar than for Sobe. -State the null hypothesis in symbols. Also clearly define what the symbols mean in this context. Consider the following ANOVA output:

Suppose that you have a $1 / 9$ probability of winning a game of solitaire, independently from game to game. -If you play repeatedly until you win, what is the probability that you lose four games and then win the fifth?

Suppose that you have a $1 / 9$ probability of winning a game of solitaire, independently from game to game. -If you play two games, what is the probability that you win at least once? State which probability rule(s) you use for this calculation.

A group of students asked participants from their school to see how many sit-ups they could perform in one minute. Subjects were then randomly assigned to drink either Sobe or Rockstar, two energy drinks. Rockstar is marketed primarily as an energy drink, whereas Sobe is advertised more for taste. Exactly 30 minutes later, the number of sit-ups in one minute was again recorded. The students conjectured that the increase in the number of sit-ups performed would be higher for students drinking Rockstar than for Sobe. -Summarize the conclusion you would draw from this study, with regard to the question of whether the brand of energy drink affects the increase in sit-up performance.

Suppose that 70% of the customers at a particular fast food restaurant order a hamburger, 50% order French fries, and 45% order both. Let H denote the event that a randomly selected customer orders a hamburger, and let F denote the event that he or she orders French fries. -Are the events $\{$ customer orders a hamburger $\}$ and $\{$ customer orders French fries $\}$ independent? Justify your answer numerically.

Suppose that you have a $1 / 9$ probability of winning a game of solitaire, independently from game to game. -If you play two games, what is the probability that you win both? State which probability rule you use for this calculation.

Data were collected on the weights for the entire roster of the 2005 Cal Poly football team. Individuals were classified as playing offense, defense, or special teams. Suppose you want to consider these data as representative of all collegiate football players and want to compare the average weight of the players in these three groups. -Use the output to determine the value of the $F$ statistic. Show your calculation.