The National Association of Realtors estimates that 23% of all homes purchased in 2004 were considered
investment properties. If a sample of 800 homes sold in 2004 is obtained what is the probability that at least 175
homes are going to be used as investment property
To find the probability that at least 175 homes are going to be used as investment property, we can use the binomial probability formula.
The binomial probability formula is given by:
P(X >= k) = 1 - P(X < k)
Where:
P(X >= k) is the probability of X being greater than or equal to k
P(X < k) is the probability of X being less than k
In this case, we need to find the probability that at least 175 homes are used as investment property, so we can use the complement rule:
P(X >= 175) = 1 - P(X < 175)
To calculate the probability, we need to know the number of trials (n) and the probability of success (p).
In this case, the number of trials (n) is 800 homes sold, and the probability of success (p) is 0.23 (since 23% of homes are considered investment properties).
Now, we can calculate the probability using a binomial distribution calculator or statistical software. Alternatively, we can use the normal approximation to the binomial distribution for large enough sample sizes.
Assuming that the sample size (n) is large enough, we can use the normal approximation to estimate the probability.
First, we calculate the mean (μ) of the binomial distribution using the formula:
μ = n * p
μ = 800 * 0.23 = 184
Next, we calculate the standard deviation (σ) of the binomial distribution using the formula:
σ = sqrt(n * p * (1 - p))
σ = sqrt(800 * 0.23 * (1 - 0.23)) = 12.85
Now, we can use the normal approximation to calculate the probability:
P(X >= 175) = 1 - P(X < 175)
To use the normal approximation, we standardize the values using the z-score formula:
z = (x - μ) / σ
where x is the number of homes used as investment property.
P(X >= 175) = 1 - P((X - μ) / σ < (175 - μ) / σ)
P(X >= 175) = 1 - P(z < (175 - μ) / σ)
P(X >= 175) = 1 - P(z < (175 - 184) / 12.85)
P(X >= 175) = 1 - P(z < -0.7004)
Finally, we can use a standard normal distribution table or a calculator to find the probability associated with the z-score -0.7004.
Note: The calculation assumes that the sample size is large enough for the normal approximation to be valid. If the sample size is small, it is better to use the binomial distribution directly.
To find the probability that at least 175 homes are going to be used as investment property, we can use the binomial probability formula:
P(X ≥ k) = (nCk) * p^k * (1-p)^(n-k)
Where:
P(X ≥ k) is the probability of getting at least k successes.
n is the total number of trials or observations.
k is the number of successful trials.
p is the probability of success in a single trial.
In this case:
n = 800 (total number of homes sold)
k = 175 (minimum number of homes used as investment property)
p = 0.23 (probability of a home being used as investment property)
Now, let's calculate the probability using the formula:
P(X ≥ 175) = (800C175) * (0.23)^175 * (0.77)^(800-175)
Please note that (800C175) means "800 choose 175," which represents the number of possible combinations.
To calculate this probability, you can use a statistical software, a calculator with a binomial probability function, or an online calculator.
Let me know if you would like me to perform the calculations for you using a specific method.