Question

As a special promotion for its 20-ounce bottles of soda, a soft drink company printed a message on the inside of each bottle cap.

Some of the caps said, "Please try again!" while others said, "You're a winner!" The company advertised the promotion with the slogan "1 in 6 wins a prize." Grayson's statistics class wonders if the comparty's claim holds true at a nearby convenience store.
To find out, all 30 students in the class go to the store and each buys one 20-ounce bottle of the soda.
Two of the students in Grayson's class got caps that say, "You're a winner!" Does this result give convincing evidence that the company's 1-in-6 claim is false?

A) Yes. If the "1 in 6 wins" claim is true, there is a 10.28% probability that two or fewer students would win a prize.
Because this outcome is not very unlikely, we have convincing evidence that the company's claim is false.

B) Yes. If the "1 in 6 wins" claim is true, there is a 89.72% probability that two or fewer students would win a prize.
Because this outcome is not very unlikely, we do not have convincing evidence that the company's claim is false.

C) No. If the "1 in 6 wins" claim is true, there is a 10.28% probability that two or fewer students would win a prize.
Because this outcome is not very unlikely, we do not have convincing evidence that the company's claim is false.

D)Yes. Only 6.67% of the students won a prize and 16.67% were supposed to win a prize.

E) No. If the "1 in 6 wins" claim is true, there is a 89.72% probability that two or fewer students would win a prize.
Because this outcome is not very unlikely, we have convincing evidence that the company's claim is false.

Answer 1

A hypothesis test using a binomial distribution can determine if the result of only 2 out of 30 students winning is statistically significant evidence against the company's claim of a '1 in 6' chance of winning.

The student's question revolves around the probabilities associated with a promotional event and whether the observed outcomes provide convincing evidence against the company's claim.

Given that a company claims that "1 in 6 wins a prize," we would expect that out of 30 students, about 5 should win a prize (since 30/6 = 5). However, only 2 students got winning caps.

To determine if this is significantly different from the company's claim, a hypothesis test can be conducted.

Using the binomial distribution, we can calculate the probability of 2 or fewer students winning a prize if the true probability of winning is indeed 1/6.

The calculated probability can then be compared to a significance level to determine whether the result is statistically significant. If the observed probability is low (typically less than 5% or 0.05), we might consider the evidence against the company's claim as 'convincing.'