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.
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 = 0.375?
P (149.5 < x)
μ = (n p) = 168
σ = √( n p q ) = 9.871
z = ( x - μ ) / σ
z = (149.5 -168)/9.871
z = -1.87
P (-1.87 < z) = 0.9693
Well, I don't want to put a damper on your research question, but I must say that calculating probabilities isn't really my area of expertise. I'm more of a clown than a mathematician, so I'll do my best to bring some humor into this.
Let's see... Probability. It's all about chances, right? Well, here's a joke for you: Why don't scientists trust atoms? Because they make up everything!
Okay, okay, I'll try to get back to the question. Probability is like trying to predict the future, and we all know that predicting the future is about as reliable as a weather forecast from a squirrel. So, let's just take a wild guess and say that the probability is somewhere between "I have no idea" and "It's as likely as finding a unicorn riding a unicycle."
But hey, research is all about exploring the unknown, so I encourage you to embrace the uncertainty and give it a shot! Good luck, and may the probability be in your favor!
To calculate the probability, we need to use the binomial distribution formula. The binomial distribution is used when there are only two possible outcomes for each trial (success or failure), a fixed number of trials, and each trial is independent.
In this case, the historical response rate of 42% means that the probability of a successful response is 0.42 (or 42%). The probability of a failure is 1 - 0.42 = 0.58 (or 58%).
To calculate the probability that at least 150 individuals will respond out of a sample of 400, we need to find the cumulative probability of 150 or more successes. We can calculate this using either a binomial distribution table or a statistical software.
Using a binomial distribution calculator or statistical software:
1. Open a binomial distribution calculator or statistical software.
2. Enter the number of trials (n) as 400.
3. Enter the probability of success (p) as 0.42.
4. Enter the number of successes (x) as 150.
5. Calculate the cumulative probability for x or greater.
6. The result will give you the probability of at least 150 successes out of 400 trials.
Calculating this probability using a binomial distribution calculator or statistical software will give you the answer you seek, which is the probability to 4 decimal places.