A market research firm conducts telephone surveys with a 44% 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?
0.9964
To solve this problem, we can use the binomial distribution. Let's break it down step by step:
1. Identify the parameters:
- The historical response rate is 44%, which can be converted to a probability of success, p = 0.44.
- The sample size is 400 telephone numbers, n = 400.
- We want to find the probability that at least 150 individuals will cooperate and respond, which means we are looking for P(X ≥ 150), where X is the number of successes.
2. Calculate the mean and standard deviation:
- The mean of a binomial distribution is given by μ = np, so μ = 400 * 0.44 = 176.
- The standard deviation of a binomial distribution is given by σ = sqrt(npq), where q = 1 - p. So σ = sqrt(400 * 0.44 * 0.56) ≈ 10.40.
3. Calculate the probability using the cumulative distribution function (CDF):
- To find P(X ≥ 150), we can find the complement P(X ≤ 149) and subtract it from 1.
- We can use the normal approximation to the binomial distribution because np and nq are both greater than 5. We can calculate the z-score as z = (x - μ) / σ, where x = 149.
- Calculate the z-score: z = (149 - 176) / 10.40 ≈ -2.60.
- Use a standard normal distribution table or a calculator to find the cumulative probability corresponding to z = -2.60. Let's say it's approximately 0.0047.
- Finally, subtract the cumulative probability from 1: P(X ≥ 150) = 1 - 0.0047 ≈ 0.9953.
So, the probability that in a new sample of 400 telephone numbers, at least 150 individuals will cooperate and respond to the questions is approximately 0.9953.