Consider the probability that fewer than 17 out of 145 people will get the flu this winter. Assume the probability that a given person will get he flu this winter is 11%. Approximate the probability using the normal distribution. Round the answer to four decimal places.
To approximate this probability using the normal distribution, we can use the formula for approximating the probability of a binomial distribution using a normal distribution:
P(X < x) ≈ P(Z < (x - μ) / σ)
Where:
- P(X < x) is the probability of getting fewer than x successes in a binomial distribution
- P(Z < (x - μ) / σ) is the probability of getting fewer than x successes in a normal distribution
- μ is the mean of the binomial distribution (number of successes in a given number of trials)
- σ is the standard deviation of the binomial distribution (sqrt(p * q * n), where p is the probability of success in a single trial, q is the probability of failure in a single trial, and n is the number of trials)
In this case, we want to find the probability of getting fewer than 17 successes (people who get the flu) out of 145 people, with a probability of success of 11% (0.11).
Mean (μ) = p * n = 0.11 * 145 = 15.95 (rounded to 2 decimal places)
Standard Deviation (σ) = sqrt(p * q * n) = sqrt(0.11 * 0.89 * 145) ≈ 3.1506 (rounded to 4 decimal places)
Now, we can calculate the probability using the normal distribution:
P(X < 17) ≈ P(Z < (17 - 15.95) / 3.1506)
Calculating this using a standard normal distribution table or a calculator that can compute normal distribution probabilities, we find:
P(Z < 0.651)
Looking up this value in the standard normal distribution table, we find the approximate probability to be 0.7422 (rounded to 4 decimal places).
Therefore, the approximate probability that fewer than 17 out of 145 people will get the flu this winter is 0.7422 (rounded to 4 decimal places).