Question

the u.s. housing market estimates that 31% of all homes purchased in 2019 were considered investment properties. if a sample of 800 homes sold in 2019 is obtained what is the probability that at least 175 homes are going to be used as investment property?

Answer 1

To determine the probability that at least 175 out of 800 homes sold in 2019 would be used as investment properties, with 31% being the expected rate, a normal approximation to the binomial distribution is used. The Z-score calculation shows that having at least 175 investment properties is almost certain since 175 is considerably lower than the expected number of 248.

Step-by-step explanation:

To find the probability that at least 175 homes out of a sample of 800 homes sold in 2019 will be used as investment property, given that 31% of all homes purchased in 2019 were considered investment properties, we can use the normal approximation to the binomial distribution because the sample size is large.

First, we need to determine the mean (μ) and standard deviation (σ) for the binomial distribution:

• Mean (μ) = n × p = 800 × 0.31 = 248
• Standard deviation (σ) = √(n × p × (1-p)) = √(800 × 0.31 × (1-0.31)) ≈ √(800 × 0.31 × 0.69) ≈ 13.65

Next, we apply the normal approximation to calculate the probability for at least 175 homes:

Z = (x - μ) / σ = (175 - 248) / 13.65 ≈ -5.35

Since the Z-score of -5.35 is very far on the left tail of the normal distribution, the probability of at least 175 homes being investment properties is nearly 1 (or almost certain) because 175 is significantly less than the expected number 248.

Answer 2

The probability of at least 175 out of 800 homes being investment properties, when 31% are expected to be so, involves binomial or normal approximation calculations. The mean is 248 and the standard deviation is approximately 13.49. The z-score of -5.45 indicates the probability is very close to 1.

Step-by-step explanation:

The student asks about the probability that at least 175 out of 800 homes purchased in 2019 will be used as investment properties given that 31% of homes are estimated to be investment properties. This probability problem can be solved using the binomial distribution or normal approximation to the binomial distribution.

To find the probability using normal approximation, first calculate the mean (μ) and standard deviation (σ) for the distribution. The mean is μ = np = 800 * 0.31 = 248. The standard deviation is σ = √np(1 - p) = √(800 * 0.31 * 0.69) ≈ 13.49. To determine the probability of at least 175 homes, we find the z-score for 174.5 (subtracting 0.5 for continuity correction) and use the standard normal distribution to find the probability for z.

The z-score calculation is (174.5 - 248) / 13.49 ≈ -5.45. Looking this z-score up in the standard normal distribution table (or using a calculator), we find that the probability of at least 175 homes being used as investment properties is very close to 1 since a z-score of -5.45 corresponds to a very small left-tail probability.