According to the Census Bureau, 64.5% of Americans owned their own home in 2013. A local real estate office wants to see if this is the case for its area. The office selects a
Random sample of 200 people to estimate the percentage who own their own homes.
What is the probability that at least 140 people own their own home?
A) 0.7291
B) 0.052
C) 0.4598
D) 0.5402
E) 0.104

Question

Quizplus · Accepted Answer

To find the probability that at least 140 out of 200 people own their own home, given that the national average is 64.5%, we can use the normal approximation to the binomial distribution. The mean ($\mu$) of the distribution is $n \times p = 200 \times 0.645 = 129$, and the standard deviation ($\sigma$) is $\sqrt{n \times p \times (1-p)} = \sqrt{200 \times 0.645 \times (1-0.645)}$. The Z-score for 140 people is calculated by $(X - \mu) / \sigma$, and then we find the probability corresponding to this Z-score using the standard normal distribution. The probability of at least 140 people owning their own home would be very low, given the national average, which aligns with option B, indicating it's a rare event.