Calculating Probabilities for Sampling Distribution Answers

A fair six-sided number cube is rolled 60 times. What is the probability that fewer than 10% of the rolls are a five?Find the z-table here.

An acting school claims that 71% of its graduates land major acting roles within one year of graduation. A random sample of 50 graduates was surveyed, and 30 of them had landed major roles within one year of graduation. Let = the proportion of the sample who had landed major roles within one year of graduation.The probability that 60% or fewer graduates land a major acting role within one year of graduating from this school is 0.043. Does this result provide convincing evidence against the school’s claim?

A producer of gel pens claims that 96% of its pens can write more than 10,000 words without losing ink. A random sample of 500 pens is collected, and it is found that 470 of the pens can write more than 10,000 words without losing ink. Let = the proportion of the sample of pens that can write more than 10,000 words.The probability that 94% or fewer of these gel pens can write more than 10,000 words is 0.0115. Does this result provide convincing evidence against the producer of the gel pens?

In a large urban high school, 68% of the students take public transportation to and from school. Annette takes a random sample of 75 students from this school. What is the probability that more than 70% of the students in the sample take public transportation to and from school?Find the z-table here.

A coin is bent so that, when tossed, "heads” appears two-thirds of the time. What is the probability that more than 70% of 100 tosses result in "heads”?Find the z-table here.

Driving instructors Mr. Adams and Mr. Bateman teach class independently of each other. Among Mr. Adams’s students, 68% pass the driving test on the first try, while 74% of Mr. Bateman’s students pass the driving test on the first try. Suppose there are 40 students in Mr. Adams’s class and 50 students in Mr. Bateman’s class. Let A = the proportion of students who pass the driving test on the first try from Mr. Adams’s class and B = the proportion of students who pass the driving test on the first try from Mr. Bateman’s class. What is the probability that Mr. Bateman’s class has more students who pass on the first try?Find the z-table here.

Ricardo and Tammy practice putting golf balls. Ricardo makes 47% of his putts and Tammy makes 51% of her putts. Suppose that Ricardo attempts 25 putts and Tammy attempts 30 putts. Let R = the proportion of putts Ricardo makes and T = the proportion of putts Tammy makes. What is the probability that Ricardo makes a higher proportion of putts than Tammy?Find the z-table here.

King, a golden retriever, can find a tennis ball hidden in his yard 89% of the time, while Tessa, a Labrador mix, can find a tennis ball hidden in her yard 94% of the time. Suppose King’s owner hides a tennis ball in his yard 100 times and Tessa’s owner hides a tennis ball in her yard 150 times. Let K = the proportion of times King finds the tennis ball and T = the proportion of times Tessa finds the tennis ball. What is the probability that the proportion of tennis balls King finds is greater than the proportion Tessa finds?Find the z-table here.

At a certain high school, 38% of freshmen and 31% of sophomores walk to school. Suppose that random samples of 40 freshmen and 45 sophomores are chosen. Let S = the proportion of sophomores who walk to school and F = the proportion of freshmen who walk to school. What is the probability that the proportion of sophomores who walk to school is greater than the proportion of freshmen?Find the z-table here.

Carol and Diane are axe throwers. Carol hits the board on 44% of her throws, while Diane hits the board on 42% of her throws. Suppose that Carol throws 25 axes at the board and Diane throws 28. Let C = the proportion of axes that hit the board when Carol throws and D = the proportion of axes that hit the board when Diane throws. What is the probability that Diane’s proportion of axes hitting the board is higher than Carol’s?Find the z-table here.