Question

Suppose 47% of the population has a college degree. If a random sample of size 780 is selected, what is the probability that the proportion of persons with a college degree will be less than 44%? Round your answer to four decimal places.

Answer 1

Answer:

0.0465 = 4.65% probability that the proportion of persons with a college degree will be less than 44%

Step-by-step explanation:

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:

$n = 780, p = 0.47$

So

$\mu = E(X) = np = 780*0.47 = 366.6$

$\sigma = \sqrt{V(X)} = \sqrt{np(1-p)} = \sqrt{780*0.47*0.53} = 13.94$

What is the probability that the proportion of persons with a college degree will be less than 44%?

This is the pvalue of Z when X = 0.44*780 = 343.2. So

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

$Z = \frac{343.2 - 366.6}{13.94}$

$Z = -1.68$

$Z = -1.68$ has a pvalue of 0.0465

0.0465 = 4.65% probability that the proportion of persons with a college degree will be less than 44%

Answer 2

Answer:

The probability that the proportion of persons with a college degree will be less than 44% is 0.0455.

Step-by-step explanation:

We are given that 47% of the population has a college degree and a random sample of size 780 is selected.

Let p = % of population who has a college degree = 47%

The z-score probability distribution for proportion is given by;

                 Z =  $\frac{\hat p-p}{\sqrt{\frac{\hat p(1-\hat p)}{n} } }$ ~ N(0,1)

where, $\hat p$  = proportion of persons with a college degree in a sample of 780 selected

            n = sample size = 780

So, probability that the proportion of persons with a college degree will be less than 44% is given by = P($\hat p$ < 0.44)

    P($\hat p$ < 0.44) = P( $\frac{\hat p-p}{\sqrt{\frac{\hat p(1-\hat p)}{n} } }$ < $\frac{0.44-0.47}{\sqrt{\frac{0.44(1-0.44)}{780} } }$ ) = P(Z < -1.69) = 1 - P(Z $\leq$ 1.69)

                                                                  = 1 - 0.95449 = 0.0455

The above probability is calculated using z table by looking at value of x = 1.69 in the z table which have an area of 0.95449.

Therefore, probability that the proportion of persons with a college degree will be less than 44% is 0.0455.