Calculating Probabilities for Sampling Distribution Answers

In a certain town, 62% of the residents enjoy riding roller coasters. Alfred takes a sample of 75 residents from this town. What is the probability that fewer than 58% of the residents in the sample enjoy riding roller coasters?Find the z-table here.

A fair six-sided number cube has the following faces: 1, 1, 2, 2, 5, 6. This number cube is rolled 50 times. What is the probability that fewer than 30% of the rolls result in a two?Find the z-table here.

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?

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?

In a beach town, 13% of the residents own boats. A random sample of 100 residents was selected. What is the probability that less than 11% of the residents in the sample own boats?Find the z-table here.

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.

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.

A ceiling fan manufacturer claims that 89% of homes in the South have ceiling fans. A random sample of 100 Southern homes was surveyed, and 85 had ceiling fans. Let = the proportion of the sample homes that have ceiling fans.The probability that 85% or fewer homes in the South have ceiling fans is 0.098. Does this result provide convincing evidence against the manufacturer’s claim?

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.