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 1000 is between 22% and 28%.
0.8866
0.9714
0.8413
0.9287
To find the probability that the proportion of successes in a sample of 1000 is between 22% and 28%, we can use the normal approximation to the binomial distribution.
First, calculate the mean (µ) and standard deviation (σ) of the binomial distribution:
µ = np = 1000 * 0.25 = 250
σ = sqrt(np(1-p)) = sqrt(1000 * 0.25 * 0.75) = 12.9099
Next, convert the proportion of successes into a z-score for both 22% and 28%, using the formula z = (x-µ)/σ:
For 22%:
z = (0.22 - 0.25) / 12.9099 = -0.0293
For 28%:
z = (0.28 - 0.25) / 12.9099 = 0.0293
Now, use a standard normal table or a calculator to find the probabilities associated with these z-scores. The probability that the proportion of successes in a sample of 1000 is between 22% and 28% is:
P(z < 0.0293) - P(z < -0.0293)
Looking up the z-scores in a standard normal table, we find the corresponding probabilities:
P(z < 0.0293) is approximately 0.5110
P(z < -0.0293) is approximately 0.4591
Therefore, the probability that the proportion of successes in a sample of 1000 is between 22% and 28% is:
0.5110 - 0.4591 = 0.0519
Therefore, the closest answer provided is 0.0521, which is probably a rounding error in calculation.