Answer:
95% CI: [0.25;0.29]
Step-by-step explanation:
Hello!
1572 randomly selected people where asked if the government should give financial assistance to college students from low-income families, 428 answered affirmatively.
You need to estimate using a 95% CI the proportion of Americans that believe it is the government's responsibility.
First, identify the study variable:
X: Number of people that think the government should give financial assistance to college students from low-income families out of 1572 surveyed Americans.
This variable has a binomial distribution and since the sample is large enough (n≥30; n*p'≥5 and n*(1-p')≥5), you can apply the Central Limit Theorem to approximate the distribution of the sampling proportion to normal and use this approximation to calculate the Confidence Interval:
p' ±
* ![\sqrt{\frac{p'(1-p')}{n} }](https://tex.z-dn.net/?f=%5Csqrt%7B%5Cfrac%7Bp%27%281-p%27%29%7D%7Bn%7D%20%7D)
sample proportion p'=428/1572= 0.27
![Z_{1-\alpha /2}= Z_{0.975}= 1.965](https://tex.z-dn.net/?f=Z_%7B1-%5Calpha%20%2F2%7D%3D%20Z_%7B0.975%7D%3D%201.965)
0.27 ± 1.965 * ![\sqrt{\frac{0.27*0.73}{1572} }](https://tex.z-dn.net/?f=%5Csqrt%7B%5Cfrac%7B0.27%2A0.73%7D%7B1572%7D%20%7D)
[0.247;0.292] ≅ [0.25;0.29]
I hope it helps!