If 15% of adults in a certain country work from home, what is the probability that fewer than 54 out of a random sample of 450 adults will work from home? (Round your final answer to 4 decimal places.)
Answer (2)
The probability that fewer than 54 out of a random sample of 450 adults will work from home is approximately 0.0323. This calculation is based on the binomial distribution's approximation using the normal distribution. The final answer is rounded to four decimal places as 0.0323.
;
Define the random variable X as the number of adults who work from home, following a binomial distribution.
Approximate the binomial distribution with a normal distribution with μ = 67.5 and σ ≈ 7.5746 .
Apply the continuity correction and calculate the z-score: z ≈ − 1.848 .
Find the probability using the standard normal distribution: P ( Z < − 1.848 ) ≈ 0.0323 .
Explanation
Understand the problem We are given that 15% of adults in a certain country work from home. We want to find the probability that fewer than 54 out of a random sample of 450 adults will work from home.
Define the random variable Let X be the number of adults who work from home in the sample of 450. Then X follows a binomial distribution with n = 450 and p = 0.15 . We want to find P ( X < 54 ) , which is the same as P ( X ≤ 53 ) .
Calculate mean and standard deviation Since n is large, we can approximate the binomial distribution with a normal distribution. The mean and standard deviation of the binomial distribution are: μ = n p = 450 × 0.15 = 67.5 σ = n p ( 1 − p ) = 450 × 0.15 × 0.85 = 57.375 ≈ 7.5746
Apply continuity correction We apply the continuity correction: P ( X ≤ 53 ) is approximated by P ( Y < 53.5 ) , where Y is a normal random variable with mean 67.5 and standard deviation 7.5746.
Calculate the z-score Calculate the z-score: z = 7.5746 53.5 − 67.5 = 7.5746 − 14 ≈ − 1.848
Find the probability Find the probability using the standard normal distribution: P ( Y < 53.5 ) = P ( Z < − 1.848 ) , where Z is a standard normal random variable. Using a z-table or calculator, we find P ( Z < − 1.848 ) ≈ 0.0323 .
Round the answer Rounding the final answer to 4 decimal places, we get 0.0323.
Examples
This type of probability calculation is useful in various real-world scenarios. For example, a marketing company might want to estimate the probability that a certain number of people in a sample will respond to an advertisement, based on the overall response rate. Similarly, a public health organization might want to estimate the probability that a certain number of people in a sample will contract a disease, based on the overall prevalence of the disease in the population. These calculations help in making informed decisions and predictions.
