If 15% of adults in a certain country work from home, what is the probability that fewer than 36 out of a random sample of 300 adults will work from home? (Round your final answer to 4 decimal places.)
Answer (2)
The probability that fewer than 36 out of 300 adults will work from home is approximately 0.0623. This calculation is based on the normal approximation of a binomial distribution. By using continuity correction, we found the z-score and looked up the corresponding probability in the standard normal distribution.
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Define the binomial distribution and approximate with a normal distribution.
Calculate the mean: μ = 45 .
Calculate the standard deviation: σ ≈ 6.1847 .
Apply continuity correction and calculate the z-score: z ≈ − 1.5361 , then find the probability: 0.0623 .
Explanation
Understand the problem We are given that 15% of adults work from home. We want to find the probability that fewer than 36 out of a random sample of 300 adults work from home. This is a binomial distribution problem that can be approximated using a normal distribution.
Define the distribution Let X be the number of adults who work from home in the sample. Then X follows a binomial distribution with n = 300 and p = 0.15 . We can approximate this binomial distribution with a normal distribution with mean μ = n p and standard deviation σ = n p ( 1 − p ) .
Calculate the mean Calculate the mean: μ = 300 × 0.15 = 45 .
Calculate the standard deviation Calculate the standard deviation: σ = 300 × 0.15 × ( 1 − 0.15 ) = 300 × 0.15 × 0.85 = 38.25 ≈ 6.1847 .
Apply continuity correction We want to find P ( X < 36 ) . Using the continuity correction, we approximate this with P ( Y < 35.5 ) , where Y is a normal random variable with mean 45 and standard deviation 6.1847.
Calculate the z-score Calculate the z-score: z = 6.1847 35.5 − 45 ≈ − 1.5361 .
Find the probability Find the probability using the standard normal distribution: P ( Z < − 1.5361 ) ≈ 0.0623 .
Round the answer Round the final answer to 4 decimal places: 0.0623 .
Examples
This type of probability calculation is useful in various real-world scenarios. For example, a company might want to estimate the likelihood that a certain number of their employees will work from home on a given day. This can help them plan for office space and resources. Similarly, public health officials could use this to predict the number of people who might need access to remote healthcare services.
