Question

A survey of 2,450 adults reported that 57​% watch news videos. Complete parts​ (a) through​ (c) below. Question content area bottom Part 1 a. Suppose that you take a sample of 50 adults. If the population proportion of adults who watch news videos is 0.57​, what is the probability that fewer than half in your sample will watch news​ videos? The probability is enter your response here that fewer than half of the adults in the sample will watch news videos. ​(Round to four decimal places as​ needed.) Part 2 b. Suppose that you take a sample of 250 adults. If the population proportion of adults who watch news videos is 0.57​, what is the probability that fewer than half in your sample will watch news​ videos? The probability is enter your response here that fewer than half of the adults in the sample will watch news videos. ​(Round to four decimal places as​ needed.) Part 3 c. Discuss the effect of sample size on the sampling distribution of the proportion in general and the effect on the probabilities in parts​ (a) and​ (b). Choose the correct answer below. A. The probabilities in parts​ (a) and​ (b) are the same. Increasing the sample size does not change the sampling distribution of the proportion. B. Increasing the sample size by a factor of 5 decreases the standard error by a factor of 5. This causes the sampling distribution of the proportion to become more concentrated around the true population proportion of 0.57 and decreases the probability in part​ (b). C. Increasing the sample size by a factor of 5 increases the standard error by a factor of 5. This causes the sa

Answer 1

Using the normal distribution and the central limit theorem, it is found that:

a) The probability is of 0.1587 = 15.87% that fewer than half of the adults in the sample will watch news videos.

b) The probability is of 0.0125 = 1.25% that fewer than half of the adults in the sample will watch news videos.

c) The standard error is inversely proportional to the square root of n, hence increasing the sample size by a factor of 5 decreases the standard error by a factor of $\sqrt{5}$, which causes the sampling distribution of the proportion to become more concentrated around the true population proportion of 0.57 and decreases the probability in part​ (b).

Normal Probability Distribution

In a normal distribution with mean $\mu$ and standard deviation $\sigma$, the z-score of a measure X is given by:

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

• It 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, which is the percentile of X.

• By the Central Limit Theorem, for a proportion p in a sample of size n, the sampling distribution of sample proportion is approximately normal with mean $\mu = p$ and standard deviation $s = \sqrt{\frac{p(1 - p)}{n}}$, as long as $np \geq 10$ and $n(1 - p) \geq 10$.

In this problem, we have that the proportion is p = 0.57.

Item a:

Sample of n = 50, hence the mean and the standard error are given by:

$\mu = p = 0.57$

$s = \sqrt{\frac{p(1 - p)}{n}} = \sqrt{\frac{0.57(0.43)}{50}} = 0.07$

The probability that fewer than half in your sample will watch news​ videos is the p-value of Z when X = 0.5, hence:

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

By the Central Limit Theorem

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

$Z = \frac{0.5 - 0.57}{0.07}$

$Z = -1$

$Z = -1$ has a p-value of 0.1587.

The probability is of 0.1587 = 15.87% that fewer than half of the adults in the sample will watch news videos.

Item b:

Sample of n = 250, hence:

$s = \sqrt{\frac{p(1 - p)}{n}} = \sqrt{\frac{0.57(0.43)}{250}} = 0.0313$

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

$Z = \frac{0.5 - 0.57}{0.0313}$

$Z = -2.24$

$Z = -2.24$ has a p-value of 0.0125.

The probability is of 0.0125 = 1.25% that fewer than half of the adults in the sample will watch news videos.

Item c:

The standard error is inversely proportional to the square root of n, hence increasing the sample size by a factor of 5 decreases the standard error by a factor of $\sqrt{5}$, which causes the sampling distribution of the proportion to become more concentrated around the true population proportion of 0.57 and decreases the probability in part​ (b).

To learn more about the normal distribution and the central limit theorem, you can take a look at https://brainly.com/question/24663213