Question

In 1996, the General Social Survey asked, "On the whole, do you think it should be the governments responsibility to give financial assistance to college students from low income families?" For this question, 428 people said that it definitely should out of 1572 randomly selected people. What is the 95% confidence interval for the proportion of all Americans that believe it is the government's responsibility? Group of answer choices (0.243, 0.301) (0.250, 0.294) (0.260, 0.283) Assumptions are not met. Can not make confidence interval.

Answer 1

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' ± $Z_{1-\alpha /2}$ * $\sqrt{\frac{p'(1-p')}{n} }$

sample proportion p'=428/1572= 0.27

$Z_{1-\alpha /2}= Z_{0.975}= 1.965$

0.27 ± 1.965 * $\sqrt{\frac{0.27*0.73}{1572} }$

[0.247;0.292] ≅ [0.25;0.29]

I hope it helps!