The probability of success on any trail of a binomial experiment is 25%. Find the probability that the proportion of successes in a sample of 800 is less than 22%.
0.0250
0.0354
0.0158
0.1203
0.1829
To solve this problem, we need to use the normal approximation to the binomial distribution since the sample size is large (n=800).
First, we need to calculate the mean and standard deviation of the binomial distribution:
mean = n*p = 800*0.25 = 200
standard deviation = sqrt(n*p*q) = sqrt(800*0.25*0.75) = sqrt(150) = 12.247
Next, we will use the Z-score formula to standardize the value of 22%:
z = (X - mean) / standard deviation
z = (176 - 200) / 12.247 = -1.9596
Now, we will look up the z-score in the standard normal distribution table to find the probability that a z-score is less than -1.9596, which is 0.0250.
Therefore, the probability that the proportion of successes in a sample of 800 is less than 22% is 0.0250.