Question

In a​ survey, people were asked whether they thought sun went around the planet Earth or vice versa. Of 1910 ​people, 241 thought the sun went around Earth. a. What proportion of people in the survey believed the sun went around​ Earth? b. Find a​ 95% confidence interval for the proportion of all people with this belief. c. Suppose a scientist said that 20​% of people in the general population believe the sun goes around Earth. Using the confidence​ interval, would you say that was​ plausible? Explain your answer. a. The proportion of people in the survey who believed the sun went around Earth is . 126. ​(Round to three decimal places as​ needed.) b. The​ 95% confidence interval is (nothing comma nothing ). ​(Round to three decimal places as​ needed.) c. Is the​ scientist's claim that 20​% of people in the general population believe the sun goes around Earth​ plausible? No, it is not ​plausible, since the value . 126 is not inside of the confidence interval. ​(Type an integer or a decimal. Do not​ round.)

Answer 1

Answer:

a) 0.1203

b) The 95% confidence interval for the proportion of all people with this belief is (0.1057, 0.1349).

c) 20% is not plausible, since it is not part of the confidence interval.

Step-by-step explanation:

We have that:

1920 asked if the sun went around the planet Earth or vice versa.

241 thought the sun went around Earth.

a. What proportion of people in the survey believed the sun went around​ Earth?

$p = \frac{241}{1920} = 0.1203$

b. Find a​ 95% confidence interval for the proportion of all people with this belief.

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 $1 - \frac{\alpha}{2}$.

For this problem, we have that:

$n = 1910, p = 0.1203$

95% confidence level

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

The lower limit of this interval is:

$\pi - z\sqrt{\frac{\pi(1-\pi)}{n}} = 0.1203 - 1.96\sqrt{\frac{0.1203*0.8797}{1910}} = 0.1057$

The upper limit of this interval is:

$\pi + z\sqrt{\frac{\pi(1-\pi)}{n}} = 0.1203 + 1.96\sqrt{\frac{0.1203*0.8797}{1910}} = 0.1349$

The 95% confidence interval for the proportion of all people with this belief is (0.1057, 0.1349).

c. Suppose a scientist said that 20​% of people in the general population believe the sun goes around Earth. Using the confidence​ interval, would you say that was​ plausible? Explain your answer.

We are 95% sure that the true proportion of the general population who believe the sun goes around Earth is between (0.1057, 0.1349)

So 20% is not plausible, since it is not part of the confidence interval.

Answer 2

Based on the data provided;

• proportion of people in the survey that believed the sun went around Earth is 0.126
• The 95% confidence interval for the proportion of all people with this belief is (0.111, 0.141).
• Since 20% is not within the confidence interval, the scientist's claim is not plausible.

What is a proportion?

A proportion is the ratio of a certain amount of a whole.

From the given data:

Of 1920 were interviewed, 241 thought the sun went around Earth.

Proportion of people in the survey that believed the sun went around Earth is p.

p = 241/1910

p = 0.126

A 95% confidence interval for the proportion of all people with this belief is calculated as follows:

number of people surveyed = n

probability of a success of p,

confidence level = 1 - α,

the confidence interval is:

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

where:

z is the z-score that has a p-value of 1 - α/2

For a 95% confidence level, α = 0.05

p-value = 1 - 0.05/2

p-value = 0.975

Z = 1.96

1 - p = 0.874

The lower limit of this interval is:

$0.126 - 1.96 \sqrt{ \frac{0.126 \times 0.874}{1910} }$

Lower limit = 0.111

The upper limit of this interval is:

$0.126 - 1.96 \sqrt{ \frac{0.126 \times 0.874}{1910} }$

Upper limit = 0.141

The 95% confidence interval for the proportion of all people with this belief is (0.111, 0.141).

We are 95% confident that the true proportion of the general population who believe that the sun goes around Earth is between (0.1057, 0.1349)

Since 20% is not within the confidence interval, the scientist's claim that 20% of people in the general population believe the sun goes around Earth is not plausible.

Learn more about proportions and confidence intervals at: https://brainly.com/question/26267323