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 home 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 normal approximation to the binomial distribution.
First, let's find the mean and the standard deviation for the binomial distribution.
Mean (μ) = n * p
Standard Deviation (σ) = √(n * p * (1 - p))
Where n is the sample size (800) and p is the proportion of investment properties (0.23).
Mean (μ) = 800 * 0.23 = 184
Standard Deviation (σ) = √(800 * 0.23 * (1 - 0.23)) = √(147.92) = 12.16
Now, we will find the z-score for the given value (175).
z = (x - μ) / σ
z = (175 - 184) / 12.16 = -9 / 12.16 = -0.74
Using a standard normal distribution table or a calculator, we find the probability of a z-score less than -0.74 to be about 0.2295.
Since we need the probability of at least 175 homes, i.e., 1 minus the probability of less than 175 homes, we have:
Probability (at least 175 homes) = 1 - 0.2295 = 0.7705
So, the probability that at least 175 homes are going to be used as an investment property is approximately 0.7705 or 77.05%.
To calculate the probability that at least 175 homes are going to be used as investment properties, we need to use the binomial distribution formula.
The binomial distribution is used when there are only two possible outcomes for each trial (in this case, a home is either an investment property or not), and the probability of success is the same for each trial.
The binomial distribution formula is:
P(X ≥ k) = 1 - P(X < k)
where:
P(X ≥ k) is the probability of getting at least k successes.
P(X < k) is the probability of getting fewer than k successes.
In this case, the probability of a home being used as an investment property is 23%, which can be written as 0.23. The number of trials is 800 (the sample size).
Now, let's calculate the probability using the formula:
P(X ≥ 175) = 1 - P(X < 175)
To calculate P(X < 175), we need to calculate the cumulative probability of getting fewer than 175 successes.
P(X < 175) = Σ [800Ck * (0.23)k * (0.77)^(800-k)] for k = 0 to 174
(800Ck represents the combination of 800 items taken k at a time, and (^) represents exponentiation here)
Since calculating the cumulative probability manually involves quite a number of calculations, it's more convenient to use software like Excel or statistical calculators. You can use the binomial distribution function in Excel or other statistical packages to find P(X < 175).
After finding P(X < 175), you can substitute the value in the formula:
P(X ≥ 175) = 1 - P(X < 175)
This will give you the probability that at least 175 homes will be used as investment properties in the sample of 800 homes sold in 2004.
To calculate the probability that at least 175 out of 800 homes sold in 2004 will be used as investment properties, you can use the binomial probability formula.
The formula for the probability of getting exactly k successes in n independent Bernoulli trials with probability p of success on each trial is:
P(X = k) = C(n, k) * p^k * (1-p)^(n-k)
Where:
P(X = k) is the probability of getting exactly k successes,
C(n, k) is the binomial coefficient (n Choose k),
p is the probability of success on each trial, and
n is the number of trials.
In this case, n = 800, the probability of success (p) is 0.23 (23%), and k is at least 175.
To find the probability of at least 175 homes being used as investment properties, you need to calculate the probability of getting 175, 176, 177, ..., up to 800.
P(X >= 175) = P(X = 175) + P(X = 176) + ... + P(X = 800)
To calculate this probability, you can use a statistical software or calculator that allows you to calculate binomial probabilities.