Final answer:
Using binomial distribution formulae, probability questions can be solved regarding the number of individuals not covering their mouths when sneezing, based on a given success probability. Calculations include finding the probability of exactly 7 out of 18 people not covering, the probability of fewer than 3 not covering, and discussing if it would be surprising if fewer than half covered their mouths.
Step-by-step explanation:
The questions posed relate to the use of the binomial distribution to solve probability questions about sneezing habits based on a given success probability.
- Exactly 7 people not covering mouths: The probability of exactly 7 out of 18 people not covering their mouths when sneezing can be calculated using the binomial probability formula: P(X=k) = (n choose k) * p^k * (1 - p)^(n - k), where P(X=k) is the probability of k successes in n trials, p is the probability of success, and (n choose k) is the binomial coefficient.
- Fewer than 3 not covering mouths: To find the probability of fewer than 3 not covering, you would add the probabilities of 0, 1, and 2 people not covering. This sum can be calculated using the same formula as in part (a).
- Surprise if fewer than half cover: Since the expectation is that a majority will cover their mouth based on the given success probability (0.267 for not covering), observing fewer than half covering would be surprising. This requires calculating the cumulative probability for k = 0 to k = 8 (since 9 is half of 18) and comparing it to a threshold for surprise, which could be a level of significance like 0.05.
The complete question is:According to a study done by Nick Wilson of Otago University Wellington, the probability a randomly selected individual will not cover his or her mouth when sneezing is 0.267. Suppose you sit on a bench in a mall and observe people's habits as they sneeze. Complete parts (a) through (c). (a) What is the probability that among 18 randomly observed individuals, exactly 7 do not cover their mouth when sneezing? Using the binomial distribution, the probability is (Round to four decimal places as needed.) (b) What is the probability that among 18 randomly observed individuals, fewer than 3 do not cover their mouth when sneezing? Using the binomial distribution, the probability is (Round to four decimal places as needed.) (c) Would you be surprised if, after observing 18 individuals, fewer than half covered their mouth when sneezing? Why? it be surprising, because using the binomial distribution, the probability is which is 0.05. cimal places as needed.) According to a study done by Nick Wilson of Otago University Wellington, the probability a randomly selected individual will not cover his or her mouth when sneezing is 0.267. Suppose you sit on a bench in a mall and observe people's habits as they sneeze. Complete parts (a) through (c). (a) What is the probability that among 18 randomly observed individuals, exactly 7 do not cover their mouth when sneezing? Using the binomial distribution, the probability is (Round to four decimal places as needed.) (b) What is the probability that among 18 randomly observed individuals, fewer than 3 do not cover their mouth when sneezing? Using the binomial distribution, the probability is 0.1039. (Round to four decimal places as needed.) (c) Would you be surprised if, after observing 18 individuals, fewer than half covered their mouth when sneezing? Why? it be surprising, because using the binomial distribution, the probability is, which is 0.05. (Round i needed.) According to a study done by Nick Wilson of Otago University Wellington, the probability a randomly selected individual will not cover his or her mouth when sneezing is 0.267. Suppose you sit on a bench in a mall and observe people's habits as they sneeze. Complete parts (a) through (c). (a) What is the probability that among 18 randomly observed individuals, exactly 7 do not cover their mouth when sneezing? Using the binomial distribution, the probability is (Round to four decimal places as needed.) (b) What is the probability that among 18 randomly observed individuals, fewer than 3 do not cover their mouth when sneezing? Using the binomial distribution, the probability is 0.1039. (Round to four decimal places as needed.) (c) Would you be surprised if, after observing 18 individuals, fewer than half covered their mouth when sneezing? Why? it be surprising, because using the binomial distribution, the probability is , which is 0.05. (Round to four decimal places as needed.) is: