Complete Question
It is found that on average, 9% of the population recycles its garbage. Under this assumption, if a random sample of 120 households in Iowa City is taken, would you be surprised to find that fewer than 3% of households in the sample recycle their garbage? Use the fact that p-hat has an approximately normal distribution.
Group of answer choices
A I would not be surprised because there are lots of lazy students in Iowa City.
B I would not be surprised because 3% is not that far away from 9%.
C I would be surprised because Iowa City is a very clean community which means lots of people recycle their garbage.
D I would be surprised because if the 9% average is true, the chance of a sample proportion being less than 3% is very small (only 1%). Who cares? It's only garbage!
Answer:
The correct option is D
Step-by-step explanation:
From the question we are told that
The sample size is n = 120
The mean of the proportion is p = 0.09
Generally the standard deviation is mathematically represented as
![\sigma = \sqrt{\frac{p(1-p)}{n} }](https://tex.z-dn.net/?f=%5Csigma%20%20%3D%20%20%5Csqrt%7B%5Cfrac%7Bp%281-p%29%7D%7Bn%7D%20%7D)
=> ![\sigma = \sqrt{\frac{0.09(1-0.09)}{120 } }](https://tex.z-dn.net/?f=%5Csigma%20%20%3D%20%20%5Csqrt%7B%5Cfrac%7B0.09%281-0.09%29%7D%7B120%20%7D%20%7D)
=> ![\sigma = 0.02612](https://tex.z-dn.net/?f=%5Csigma%20%20%3D%200.02612)
Generally the chance that fewer than 3% of households in the sample recycle their garbage is mathematically represented as
![P(X < 0.03) = P( \frac{X - p}{\sigma} < \frac{0.03 -0.09}{0.02612} )](https://tex.z-dn.net/?f=P%28X%20%3C%200.03%29%20%3D%20%20P%28%20%5Cfrac%7BX%20-%20p%7D%7B%5Csigma%7D%20%20%3C%20%20%5Cfrac%7B0.03%20-0.09%7D%7B0.02612%7D%20%29)
![P(X < 0.03) = P( Z < -2.297 )](https://tex.z-dn.net/?f=P%28X%20%3C%200.03%29%20%3D%20%20P%28%20Z%20%20%3C%20-2.297%20%29)
From the z-table
![P( Z < -2.297 ) = 0.01](https://tex.z-dn.net/?f=P%28%20Z%20%20%3C%20-2.297%20%29%20%3D%200.01)
=> [tex]P(X < 0.03) = 0.01/tex]
=> [tex]P(X < 0.03) = 1\% /tex]