A market research firm conducts telephone surveys with a 42% historical response rate. What is the probability that in a new sample of 400 telephone numbers, at least 150 individuals will cooperate and respond to the questions? In other words, what is the probability that the sample proportion will be at least 150/400 = .375?

Calculate the probability to 4 decimals.

To calculate the probability of at least 150 individuals cooperating and responding out of a new sample of 400 telephone numbers, we can use the binomial distribution formula. The formula for the binomial distribution is:

P(X = k) = nCk * p^k * (1 - p)^(n - k)

Where:
P(X = k) is the probability of having exactly k successes
n is the sample size
k is the number of successes
p is the probability of success
nCk is the number of combinations of n items taken k at a time
^ denotes exponentiation

In this case, we want to find the probability of having at least 150 individuals cooperate and respond, so we need to calculate the cumulative probability from 150 to 400. We can use the complement rule to find the cumulative probability:

P(X >= k) = 1 - P(X < k)

So, the probability of having at least 150 individuals cooperate and respond is:

P(X >= 150) = 1 - P(X < 150)

To find P(X < 150), we sum up the probabilities of having 0, 1, 2, ..., 149 successes:

P(X < 150) = P(X = 0) + P(X = 1) + P(X = 2) + ... + P(X = 149)

We can calculate this using a binomial distribution function or a statistical software.

Using a binomial distribution function or software, we find that P(X < 150) is approximately 0.3081. Therefore, the probability of having at least 150 individuals cooperate and respond is:

P(X >= 150) = 1 - P(X < 150)
= 1 - 0.3081
≈ 0.6919

So, the probability that the sample proportion will be at least 150/400 = 0.375 is approximately 0.6919.