Answer:
a)
, in which z is related to the confidence level.
b) A sample size of 991 is needed.
Step-by-step explanation:
In a sample with a number n of people surveyed with a probability of a success of
, and a confidence level of
, we have the following confidence interval of proportions.
![\pi \pm z\sqrt{\frac{\pi(1-\pi)}{n}}](https://tex.z-dn.net/?f=%5Cpi%20%5Cpm%20z%5Csqrt%7B%5Cfrac%7B%5Cpi%281-%5Cpi%29%7D%7Bn%7D%7D)
In which
z is the zscore that has a pvalue of
.
The margin of error is:
![M = z\sqrt{\frac{\pi(1-\pi)}{n}}](https://tex.z-dn.net/?f=M%20%3D%20z%5Csqrt%7B%5Cfrac%7B%5Cpi%281-%5Cpi%29%7D%7Bn%7D%7D)
In 16% of all homes with a stay-at-home parent, the father is the stay-at-home parent
This means that ![\pi = 0.16](https://tex.z-dn.net/?f=%5Cpi%20%3D%200.16)
a. What sample size is needed if the research firm's goal is to estimate the current proportion of homes with a stay-at-home parent in which the father is the stay-at-home parent with a margin of error of 0.03 (round up to the next whole number).
This is n for which
. So
![M = z\sqrt{\frac{\pi(1-\pi)}{n}}](https://tex.z-dn.net/?f=M%20%3D%20z%5Csqrt%7B%5Cfrac%7B%5Cpi%281-%5Cpi%29%7D%7Bn%7D%7D)
![0.03 = z\sqrt{\frac{0.16*0.84}{n}}](https://tex.z-dn.net/?f=0.03%20%3D%20z%5Csqrt%7B%5Cfrac%7B0.16%2A0.84%7D%7Bn%7D%7D)
![0.03\sqrt{n} = z\sqrt{0.16*0.84}](https://tex.z-dn.net/?f=0.03%5Csqrt%7Bn%7D%20%3D%20z%5Csqrt%7B0.16%2A0.84%7D)
![\sqrt{n} = \frac{z\sqrt{0.16*0.84}}{0.03}](https://tex.z-dn.net/?f=%5Csqrt%7Bn%7D%20%3D%20%5Cfrac%7Bz%5Csqrt%7B0.16%2A0.84%7D%7D%7B0.03%7D)
![(\sqrt{n})^2 = (\frac{z\sqrt{0.16*0.84}}{0.03})^2](https://tex.z-dn.net/?f=%28%5Csqrt%7Bn%7D%29%5E2%20%3D%20%28%5Cfrac%7Bz%5Csqrt%7B0.16%2A0.84%7D%7D%7B0.03%7D%29%5E2)
, in which z is related to the confidence level.
Question b:
99% confidence level,
So
, z is the value of Z that has a pvalue of
, so
.
![n = (\frac{z\sqrt{0.16*0.84}}{0.03})^2](https://tex.z-dn.net/?f=n%20%3D%20%28%5Cfrac%7Bz%5Csqrt%7B0.16%2A0.84%7D%7D%7B0.03%7D%29%5E2)
![n = (\frac{2.575\sqrt{0.16*0.84}}{0.03})^2](https://tex.z-dn.net/?f=n%20%3D%20%28%5Cfrac%7B2.575%5Csqrt%7B0.16%2A0.84%7D%7D%7B0.03%7D%29%5E2)
![n = 990.2](https://tex.z-dn.net/?f=n%20%3D%20990.2)
Rounding up
A sample size of 991 is needed.
Answer:
5
Step-by-step explanation:
Answer:
There is no solution
Step-by-step explanation:
1. y = 3/4x - 5
2. -3x + 4y = -20
-3x + 4(3/4x - 5) = -20
-3x + 3x - 20 = - 20
-20 = -20