Using the <u>normal distribution and the central limit theorem</u>, it is found that there is a 0.0166 = 1.66% probability of a sample proportion of 0.59 or less.
In a normal distribution with mean
and standard deviation
, the z-score of a measure X is given by:
- 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, the sampling distribution of sampling proportions of a proportion p in a sample of size n has mean
and standard error ![s = \sqrt{\frac{p(1 - p)}{n}}](https://tex.z-dn.net/?f=s%20%3D%20%5Csqrt%7B%5Cfrac%7Bp%281%20-%20p%29%7D%7Bn%7D%7D)
In this problem:
- 1,190 adults were asked, hence
![n = 1190](https://tex.z-dn.net/?f=n%20%3D%201190)
- In fact 62% of all adults favor balancing the budget over cutting taxes, hence
.
The mean and the standard error are given by:
![\mu = p = 0.62](https://tex.z-dn.net/?f=%5Cmu%20%3D%20p%20%3D%200.62)
![s = \sqrt{\frac{p(1 - p)}{n}} = \sqrt{\frac{0.62(0.38)}{1190}} = 0.0141](https://tex.z-dn.net/?f=s%20%3D%20%5Csqrt%7B%5Cfrac%7Bp%281%20-%20p%29%7D%7Bn%7D%7D%20%3D%20%5Csqrt%7B%5Cfrac%7B0.62%280.38%29%7D%7B1190%7D%7D%20%3D%200.0141)
The probability of a sample proportion of 0.59 or less is the <u>p-value of Z when X = 0.59</u>, hence:
![Z = \frac{X - \mu}{\sigma}](https://tex.z-dn.net/?f=Z%20%3D%20%5Cfrac%7BX%20-%20%5Cmu%7D%7B%5Csigma%7D)
By the Central Limit Theorem
![Z = \frac{X - \mu}{s}](https://tex.z-dn.net/?f=Z%20%3D%20%5Cfrac%7BX%20-%20%5Cmu%7D%7Bs%7D)
![Z = \frac{0.59 - 0.62}{0.0141}](https://tex.z-dn.net/?f=Z%20%3D%20%5Cfrac%7B0.59%20-%200.62%7D%7B0.0141%7D)
![Z = -2.13](https://tex.z-dn.net/?f=Z%20%3D%20-2.13)
has a p-value of 0.0166.
0.0166 = 1.66% probability of a sample proportion of 0.59 or less.
You can learn more about the <u>normal distribution and the central limit theorem</u> at brainly.com/question/24663213
21/100 is a fraction form
400,000,014,104
should be your answer
hope this helps
It’s A, as you can tell from my picture
I think the first one is a simple as it gets
for the second one it could be simplified to 6x^6-4 because u combine the x terms and distribute the power