Question

A telephone poll of 1,000 adult Americans was reported in an issue of Time Magazine. One of the questions asked was “What is the main problem facing the country?” Twenty percent answered crime if the proportion of adult americans who feel that crime is the main problem is 22 percent what is the probility of getting a sample statistics like the telephone polls or higher

Answer 1

Using the normal distribution and the central limit theorem, it is found that there is a 0.937 = 93.7% probability of getting a sample statistics like the telephone polls or higher.

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 p = 0.22, n = 1000, hence the mean and the standard error are given by:

$\mu = p = 0.22$

$s = \sqrt{\frac{p(1 - p)}{n}} = \sqrt{\frac{0.22(0.78)}{1000}} = 0.0131$

The probability of getting a sample statistics like the telephone polls or higher is one subtracted by the p-value of Z when X = 0.2, hence:

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

By the Central Limit Theorem:

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

$Z = \frac{0.2 - 0.22}{0.0131}$

Z = -1.53

Z = -1.53 has a p-value of 0.063.

1 - 0.063 = 0.937.

0.937 = 93.7% probability of getting a sample statistics like the telephone polls or higher.

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