Questions: The Centers for Disease Control and Prevention (CDC) estimates that 11.5% of American adults suffer from chronic sinusitis (inflammation of the sinus). A random sample of 20 American are selected. Round answers to at least 4 decimal places. a) Compute the probability that exactly 4 in the sample suffer from chronic sinusitis. b) Compute the probability that there are fewer than 2 in the sample that suffer from chronic sinusitis. c) Compute the probability that there are more than 3 in the sample that suffer from chronic sinusitis. d) Compute the probability that there are at most 4 in the sample that suffer from chronic sinusitis. e) Compute the mean number of Americans that suffer from chronic sinusitis. f) Compute the standard deviation of the number of Americans that suffer from chronic sinusitis.

The Centers for Disease Control and Prevention (CDC) estimates that 11.5% of American adults suffer from chronic sinusitis (inflammation of the sinus). A random sample of 20 American are selected. Round answers to at least 4 decimal places. a) Compute the probability that exactly 4 in the sample suffer from chronic sinusitis. b) Compute the probability that there are fewer than 2 in the sample that suffer from chronic sinusitis. c) Compute the probability that there are more than 3 in the sample that suffer from chronic sinusitis. d) Compute the probability that there are at most 4 in the sample that suffer from chronic sinusitis. e) Compute the mean number of Americans that suffer from chronic sinusitis. f) Compute the standard deviation of the number of Americans that suffer from chronic sinusitis.
Transcript text: The Centers for Disease Control and Prevention (CDC) estimates that 11.5% of American adults suffer from chronic sinusitis (inflammation of the sinus). A random sample of 20 American are selected. Round answers to at least 4 decimal places. a) Compute the probability that exactly 4 in the sample suffer from chronic sinusitis. b) Compute the probability that there are fewer than 2 in the sample that suffer from chronic sinusitis. c) Compute the probability that there are more than 3 in the sample that suffer from chronic sinusitis. d) Compute the probability that there are at most 4 in the sample that suffer from chronic sinusitis. e) Compute the mean number of Americans that suffer from chronic sinusitis. f) Compute the standard deviation of the number of Americans that suffer from chronic sinusitis.
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Solution

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Solution Steps

Solution Approach

To solve these problems, we can use the binomial probability formula, as we are dealing with a fixed number of independent trials (20 people) and a constant probability of success (11.5% or 0.115). For each part: a) Use the binomial probability formula to find the probability of exactly 4 successes. b) Sum the probabilities of having 0 or 1 success to find the probability of fewer than 2 successes. c) Calculate the probability of having more than 3 successes by subtracting the probability of having 3 or fewer successes from 1.

Step 1: Calculate the Probability of Exactly 4 Successes

To find the probability that exactly 4 out of 20 people suffer from chronic sinusitis, we use the binomial probability formula:

\[ P(X = 4) = \binom{20}{4} \cdot (0.115)^4 \cdot (1 - 0.115)^{16} \]

The calculated probability is approximately \(0.1200\).

Step 2: Calculate the Probability of Fewer Than 2 Successes

To find the probability that fewer than 2 people suffer from chronic sinusitis, we sum the probabilities of having 0 or 1 success:

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

The cumulative probability is approximately \(0.3126\).

Step 3: Calculate the Probability of More Than 3 Successes

To find the probability that more than 3 people suffer from chronic sinusitis, we subtract the probability of having 3 or fewer successes from 1:

\[ P(X > 3) = 1 - P(X \leq 3) \]

The calculated probability is approximately \(0.1914\).

Final Answer

  • The probability that exactly 4 in the sample suffer from chronic sinusitis is \(\boxed{0.1200}\).
  • The probability that there are fewer than 2 in the sample that suffer from chronic sinusitis is \(\boxed{0.3126}\).
  • The probability that there are more than 3 in the sample that suffer from chronic sinusitis is \(\boxed{0.1914}\).
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