A typical question found on psychology exams is answered by only by 22% of the respondents. Everyone else just leaves it blank. Estimate the probability that among the next 150 responses there will be at most 40 that answer it correctly. Incidentally, the question is "Why?". Bet you don't know the answer.....
To solve this problem, we can use a binomial distribution. Let's break it down step by step:
1. Identify the variables:
- p: the probability of an individual answering the question correctly (22% or 0.22 in this case)
- n: the total number of responses (150 in this case)
- x: the number of correct responses we want to find the probability for (at most 40 in this case)
2. Calculate the probability:
We need to calculate the probability of having 0, 1, 2, ..., 40 correct responses and sum them up.
P(X ≤ 40) = P(X=0) + P(X=1) + P(X=2) + ... + P(X=40)
P(X=k) can be calculated as:
P(X=k) = (n choose k) * p^k * (1-p)^(n-k)
(n choose k) represents the binomial coefficient and can be calculated as: (n!)/(k!(n-k)!)
3. Calculate the probability for each possible value of k and sum them up:
P(X ≤ 40) = P(X=0) + P(X=1) + P(X=2) + ... + P(X=40)
You can use a calculator, statistical software, or programming language to perform the calculations. For simplicity, I will use Python to show you how to estimate the probability.
```python
import math
p = 0.22
n = 150
probability = 0
for k in range(0, 41):
binomial_coefficient = math.comb(n, k)
probability_k = binomial_coefficient * (p ** k) * ((1 - p) ** (n - k))
probability += probability_k
print("The estimated probability of having at most 40 correct responses: ", probability)
```
When you run this code, you will get the estimated probability.