Question

Suppose that you take a sample of 250 adults. If the population proportion of adults who watch news videos is 0.58​, what is the probability that fewer than half in your sample will watch news​ videos

Answer 1

Answer:

0.36% probability that fewer than half in your sample will watch news​ videos

Step-by-step explanation:

To solve this question, i am going to use the binomial approximation 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:

$n = 250, p = 0.58$

So

$\mu = E(X) = 250*0.58 = 145$

$\sigma = \sqrt{V(X)} = \sqrt{np(1-p)} = \sqrt{250*0.58*0.42} = 7.80$

If the population proportion of adults who watch news videos is 0.58​, what is the probability that fewer than half in your sample will watch news​ videos

Half: 0.5*250 = 125

Less than half is 124 or less

So this is the pvalue of Z when X = 124

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

$Z = \frac{124 - 145}{7.8}$

$Z = -2.69$

$Z = -2.69$ has a pvalue of 0.0036

0.36% probability that fewer than half in your sample will watch news​ videos

Answer 2

Answer:

Probability that fewer than half in your sample will watch news​ videos is 0.006.

Step-by-step explanation:

We are given that a sample of 250 adults is taken and the population proportion of adults who watch news videos is 0.58​.

Firstly, Let $\hat p$ = proportion of adults who watch news videos in a sample of 250 adults.

Assuming the data follows normal distribution; so the z score probability distribution for sample proportion is given by;

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

where, $p$ = population proportion of adults who watch news videos = 0.58

            n = sample of adults = 250

Probability that fewer than half in your sample will watch news​ videos is given by = P($\hat p$ < 0.50)  {As 125/250 = 0.50}

      P($\hat p$ < 125) = P( $\frac{\hat p - p}{\sqrt{\frac{\hat p(1- \hat p)}{n} } }$ < $\frac{0.50 - 0.58}{\sqrt{\frac{0.50(1-0.50)}{250} } }$ ) = P(Z < -2.53) = 1 - P(Z $\leq$ 2.53)

                                                                  = 1 - 0.9943 = 0.0057

Therefore, Probability that fewer than half in your sample will watch news​ videos is 0.006 .