Respuesta :
Answer:
a) 0.16
b) 0.0518
c) [tex]P(p \leq 0.15) = 0.4247[/tex]
Step-by-step explanation:
Problems of normally distributed samples are solved using the z-score formula.
In a set with mean [tex]\mu[/tex] and standard deviation [tex]\sigma[/tex], the zscore of a measure X is given by:
[tex]Z = \frac{X - \mu}{\sigma}[/tex]
The Z-score 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. This p-value is the probability that the value of the measure is smaller than X, that is, the percentile of X. Subtracting 1 by the pvalue, we get the probability that the value of the measure is greater than X.
For a proportion p in a sample of size n, we have that the mean is [tex]\mu = p[/tex] and the standard deviation is [tex]\sigma = \sqrt{\frac{p(1-p)}{n}}[/tex]
In this problem, we have that:
[tex]p = 0.16, n = 50[/tex]
a. Find the mean of p, where p is the proportion of minority member applications in a random sample of 2100 that is drawn from all applications.
The mean of p is 0.16.
b. Find the standard deviation of p.
[tex]\sigma = \sqrt{\frac{0.16*0.84}{50}} = 0.0518[/tex]
c. Compute an approximation for P ( p leq 0.15), which is the probability that there will be 15% or fewer minority member applications in a random sample of 2100 drawn from all applications. Round your answer to four decimal places.
This is the pvalue of Z when X = 0.15. So
[tex]Z = \frac{X - \mu}{\sigma}[/tex]
[tex]Z = \frac{0.15 - 0.16}{0.0518}[/tex]
[tex]Z = -0.19[/tex]
[tex]Z = -0.19[/tex] has a pvalue of 0.4247
[tex]P(p \leq 0.15) = 0.4247[/tex]
