Questions: 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 10 randomly observed individuals, exactly 5 do not cover their mouth when sneezing? Using the binomial distribution, the probability is 0.0724 . (Round to tour decimal places as needed.) (b) What is the probability that among 10 randomly observed individuals, fewer than 6 do not cover their mouth when sneezing? Using the binomial distribution, the probability is 0.9728 . (Round to tour decimal places as needed.) (c) Would you be surprised it, atter observing 10 individuals, fewer than half covered their mouth when sneezing? Why? Yes, it would be surprising, because using the binomial distribution, the probability is which is less than 0.05 (Round to tour decimal places as needed.)

Solution Steps

To find the probability that exactly 5 out of 10 randomly observed individuals do not cover their mouth when sneezing, we use the binomial probability formula:

\[ P(X = x) = \binom{n}{x} \cdot p^x \cdot q^{n-x} \]

where:

• \( n = 10 \) (number of trials),
• \( x = 5 \) (number of successes),
• \( p = 0.267 \) (probability of not covering mouth),
• \( q = 1 - p = 0.733 \) (probability of covering mouth).

Calculating this gives:

\[ P(X = 5) = \binom{10}{5} \cdot (0.267)^5 \cdot (0.733)^{5} \approx 0.0724 \]

Thus, the probability that exactly 5 individuals do not cover their mouth is:

\[ \boxed{0.0724} \]

To find the probability that fewer than 6 individuals do not cover their mouth, we sum the probabilities from 0 to 5:

\[ P(X < 6) = P(X = 0) + P(X = 1) + P(X = 2) + P(X = 3) + P(X = 4) + P(X = 5) \]

Calculating these probabilities, we find:

\[ P(X < 6) \approx 0.9728 \]

Thus, the probability that fewer than 6 individuals do not cover their mouth is:

\[ \boxed{0.9728} \]

To determine the probability that fewer than half (i.e., fewer than 5) of the individuals do not cover their mouth, we calculate:

\[ P(X < 5) = P(X = 0) + P(X = 1) + P(X = 2) + P(X = 3) + P(X = 4) \]

Calculating these probabilities gives:

\[ P(X < 5) \approx 0.9004 \]

Since \( 0.9004 \) is significantly greater than \( 0.05 \), it would not be surprising to observe that fewer than half of the individuals do not cover their mouth when sneezing.

Thus, the probability that fewer than half do not cover their mouth is:

\[ \boxed{0.9004} \]

Final Answer