Respuesta :
Answer:
Probability = 0.77488
Step-by-step explanation:
Since a sample was obtained, this problem is working with the sampling distribution of the sample proportion (p').l which we will need to find the probability.
A sampling distribution of the sample proportion has a mean equal to the population proportion (p).
The standard deviation of the distribution is equal to √((p(1-p))/n).
The sampling distribution in this problem will be normal because the sample size is large.
For this question, the sample proportion(p) is; 23% = 0.23
The sample size(n) is 800, so the standard deviation is;
SD = √((p(1-p))/n) = √((0.23(1-0.23))/800)
SD = 0.0149
The sample proportion is;
(p') = 175/800
p' = 0.21875
Now, for the Z-score;
z = (p' - mean)/SD
z = (0.21875 - 0.23)/ 0.0149
z = - 0.755
Now for the probability, since at least 175,it will be right tail. Thus from the table i attached;
P(z > -0.755) = 1 - 0.22663 = 0.77488
Using the normal approximation to the binomial, it is found that there is a 0.7881 = 78.81% probability that at least 175 homes are going to be used as investment property.
In a normal distribution with mean [tex]\mu[/tex] and standard deviation [tex]\sigma[/tex], the z-score of a measure X is given by:
[tex]Z = \frac{X - \mu}{\sigma}[/tex]
- It measures how many standard deviations the measure is from the mean.
- After finding the z-score, we look at the z-score table and find the p-value associated with this z-score, which is the percentile of X.
- The binomial distribution is the probability of x successes on n trials, with p probability of a success on each trial. It can be approximated to the normal distribution with [tex]\mu = np, \sigma = \sqrt{np(1-p)}[/tex].
In this problem:
- 23% of all homes purchased last year were considered investment properties, hence [tex]p = 0.23[/tex]
- Sample of 800 homes, hence [tex]n = 800[/tex].
The mean and the standard deviation for the approximation are given by:
[tex]\mu = np = 800(0.23) = 184[/tex]
[tex]\sigma = \sqrt{np(1 - p)} = \sqrt{800(0.23)(0.77)} = 11.9[/tex]
Using continuity correction, the probability that at least 175 homes are going to be used as investment property is [tex]P(X \geq 175 - 0.5) = P(X \geq 174.5)[/tex], which is 1 subtracted by the p-value of Z when X = 174.5.
[tex]Z = \frac{X - \mu}{\sigma}[/tex]
[tex]Z = \frac{174.5 - 184}{11.9}[/tex]
[tex]Z = -0.8[/tex]
[tex]Z = -0.8[/tex] has a p-value of 0.2119.
1 - 0.2119 = 0.7881
0.7881 = 78.81% probability that at least 175 homes are going to be used as investment property.
A similar problem is given at https://brainly.com/question/24261244
