Mathematics, 14.02.2020 00:13 frandariusscott

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

Answers: 2

Show answers

Answers

Answer from: lucywood2024

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

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

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

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

Answer from: Quest

hope this !

step-by-step explanation:

d = distance between moon and sun, which in this problem is the hypotenuse of a right triangle.

the distance y is the side opposite to angle x in the right triangle. use the definition of the sine function from basic trigonometry.

sin(x) = y/d ⇒

d = y/sin(x)

Hurry i need your (50 points) the moon forms a right triangle with the earth and the sun during one

Answer from: Quest

Iwould like to think so. what’s the question you need with?

Another question on Mathematics

Mathematics, 21.06.2019 17:30

What term makes it inconsistent y=2x - 4 ?

Answers: 3

Answer

Mathematics, 21.06.2019 19:00

The focus of parabola is (-4, -5), and its directrix is y= -1. fill in the missing terms and signs in parabolas equation in standard form

Answers: 1

Answer

Mathematics, 22.06.2019 00:00

Which diagram shows lines that must be parallel lines cut by transversal?

Answers: 3

Answer

Mathematics, 22.06.2019 01:50

Twelve different video games showing substance use were observed and the duration times of game play (in seconds) are listed below. the design of the study justifies the assumption that the sample can be treated as a simple random sample. use the data to construct a 99% confidence interval estimate of μ, the mean duration of game play. 4028 4316 4816 4650 4036 5010 4833 4331 4307

Answers: 1

Answer

You know the right answer?

In a local university, 40% of the students live in the dormitories. A random sample of 80 students i...

Questions