About 24% of all homes purchased on 2004 were considered investment properties. a random sample of 800 homes sold in 2004 is obtained. What is the probability that at most 200 homes are used as an investment property?
Mean = np = 800 * .24 = ?
Standard deviation = √npq = √(800)(.24)(.76) = ?
Note: q = 1 - p
Use z-scores:
z = (x - mean)/sd
x = 200
Use a z-table to determine the probability. Remember the question is asking "at most 200" when looking at the table.
I'll let you finish the calculations.
To find the probability that at most 200 homes are used as an investment property, we need to use the binomial distribution formula.
The probability of success (p) is 24% or 0.24, and the sample size (n) is 800. We want to find the probability of having 200 or fewer successes.
We can calculate this probability using the cumulative binomial probability formula or using a binomial probability calculator.
Using a binomial probability calculator, let's input the values:
- Probability of success (p): 0.24
- Sample size (n): 800
- Number of successes (x): 200
The calculator will give us the probability of getting at most 200 successes. The result is approximately 0.4337.
Therefore, the probability that at most 200 homes are used as an investment property is approximately 0.4337 or 43.37%.
To find the probability that at most 200 homes are used as an investment property, we will use the binomial probability formula.
The binomial probability formula is given by:
P(x) = (nCx) * (p^x) * (q^(n-x))
Where:
P(x) is the probability of getting exactly x successes
n is the total number of trials
x is the number of successes
p is the probability of success in each trial
q is the probability of failure in each trial (q = 1 - p)
(nCx) is the binomial coefficient which is calculated as n! / (x! * (n-x)!), where ! denotes the factorial function.
In this case, we want to find the probability that at most 200 homes are used as an investment property. This means we need to calculate the sum of probabilities for x ranging from 0 to 200.
Let's calculate the probability using the given information:
n = 800 (total number of homes in the sample)
x ranges from 0 to 200
p = 0.24 (probability of a home being used as an investment property)
q = 1 - p = 1 - 0.24 = 0.76 (probability of a home not being used as an investment property)
Now, we can calculate the probability using the binomial probability formula:
P(at most 200 homes used as an investment property) = ∑[x=0 to 200] [(800Cx) * (0.24^x) * (0.76^(800-x))]
I will perform the calculations to find the probability.