Given a random sample size of nequals900 from a binomial probability distribution with Pequals0.20, complete parts a. through e. below. Find the probability that the number of successes is fewer than 170.
To find the probability that the number of successes is fewer than 170, we can use the cumulative distribution function of the binomial distribution.
a. Identify the values of n, p, q, and x.
n = 900 (sample size)
p = 0.20 (probability of success)
q = 1 - p = 0.80 (probability of failure)
x = 169 (number of successes)
b. Calculate the mean and standard deviation of the distribution.
Mean (μ) = n * p = 900 * 0.20 = 180
Standard deviation (σ) = sqrt(n * p * q) = sqrt(900 * 0.20 * 0.80) = sqrt(144) = 12
c. Calculate the z-score for x = 169.
z = (x - μ) / σ = (169 - 180) / 12 = -0.9167
d. Look up the z-score in the standard normal table to find the corresponding cumulative probability.
Using the standard normal table or calculator, the probability associated with a z-score of -0.9167 is approximately 0.1802.
e. Calculate the probability that the number of successes is fewer than 170.
P(X < 170) = P(X ≤ 169) = 0.1802
Therefore, the probability that the number of successes is fewer than 170 in the given binomial distribution is approximately 0.1802.