Question

20. Jorge conducted a survey by asking 120 students whether they use a pencil to take notes. Of the students surveyed, 90 responded favorably. Which margin of error matches each confidence level for this situation?
1. 90% =
2. 99% =
3. 95%=
options- 0.102, 0.338, 0.216, 0.077, 0.256, 0.065

Answer 1

Answer:

Margin of error for 90% C.I: 0.065

Margin of error for 99% C.I: 0.091

Margin of error for 95% C.I: 0.077

Step-by-step explanation:

In this question, for each student, there are only two possible outcomes. Either they use a pencil to take notes, or they do not. Students are independent of each other. So we use the binomial probability distribution to solve this question.

Binomial confidence interval:

In a sample with a number n of people surveyed with a probability of a success of $\pi$, and a confidence level of $1-\alpha$, we have the following confidence interval of proportions.

$\pi \pm z\sqrt{\frac{\pi(1-\pi)}{n}}$

In which

z is the zscore that has a pvalue of $1 - \frac{\alpha}{2}$.

The margin of error is:

$M = z\sqrt{\frac{\pi(1-\pi)}{n}}$

For this problem, we have that:

120 students, so $n = 120$

90 responded favorably, so $p = \frac{90}{120} = 0.75$

Margin of error for 90% CI:

Here $\alpha = 0.1$, z is the value of Z that has a pvalue of $1 - \frac{0.1}{2} = 0.95$, so $Z = 1.645$.

The margin of error is:

$M = z\sqrt{\frac{\pi(1-\pi)}{n}} = 1.645\sqrt{\frac{0.75*0.25}{120} = 0.065$

Margin of error for 99% CI:

Here $\alpha = 0.01$, z is the value of Z that has a pvalue of $1 - \frac{0.01}{2} = 0.995$, so $Z = 2.325$.

The margin of error is:

$M = z\sqrt{\frac{\pi(1-\pi)}{n}} = 2.325\sqrt{\frac{0.75*0.25}{120} = 0.091$

Margin of error for 95% CI:

Here $\alpha = 0.05$, z is the value of Z that has a pvalue of $1 - \frac{0.05}{2} = 0.975$, so $Z = 1.96$.

The margin of error is:

$M = z\sqrt{\frac{\pi(1-\pi)}{n}} = 1.96\sqrt{\frac{0.75*0.25}{120} = 0.077$

Answer 2

Answer: Margin of error for 99% CI: z=2.575 . input that into the equation and it give you .1017

Step-by-step explanation: