Question

In a local university, 40% of the students live in the dormitories. A random sample of 80 students is selected for a particular study. The probability that the sample proportion (the proportion living in the dormitories) is between 0.30 and 0.50 is
a. 0.9328
b. 0.0336
c. 0.0672
d. 0.4664

Answer 1

Answer:

a. 0.9328

Step-by-step explanation:

For each student, either they live in the dorms, or they do not. So we use the binomial probability distribution to solve this problem.

However, we are working with samples that are considerably big. So i am going to aproximate this binomial distribution to the normal.

Binomial probability distribution

Probability of exactly x sucesses on n repeated trials, with p probability.

Can be approximated to a normal distribution, using the expected value and the standard deviation.

The expected value of the binomial distribution is:

$E(X) = np$

The standard deviation of the binomial distribution is:

$\sqrt{V(X)} = \sqrt{np(1-p)}$

Normal probability distribution

Problems of normally distributed samples can be solved using the z-score formula.

In a set with mean $\mu$ and standard deviation $\sigma$, the zscore of a measure X is given by:

$Z = \frac{X - \mu}{\sigma}$

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.

When we are approximating a binomial distribution to a normal one, we have that $\mu = E(X)$, $\sigma = \sqrt{V(X)}$.

In this problem, we have that:

$p = 0.4, n = 80$

So

$\mu = E(X) = np = 80*0.4 = 32$

$\sigma = \sqrt{V(X)} = \sqrt{np(1-p)} = \sqrt{80*0.4*0.6} = 4.3818$

The probability that the sample proportion (the proportion living in the dormitories) is between 0.30 and 0.50 is

0.30 of 80 is 0.30*80 = 24

0.50 of 80 is 0.5*80 = 40

So this probability is the pvalue of Z when X = 40 subtracted by the pvalue of Z when X = 24

X = 40

$Z = \frac{X - \mu}{\sigma}$

$Z = \frac{40 - 32}{4.3818}$

$Z = 1.83$

$Z = 1.83$ has a pvalue of 0.9664

X = 24

$Z = \frac{X - \mu}{\sigma}$

$Z = \frac{24 - 32}{4.3818}$

$Z = -1.83$

$Z = -1.83$ has a pvalue of 0.0336.

0.9664 - 0.0336 = 0.9328

So the correct answer is:

a. 0.9328