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A poll asked college students in 2016 and again in 2017 whether they believed the First Amendment guarantee of freedom of the press was secure or threatened in the country today. In 2016, 2463 of 3091 students surveyed said that freedom of the press was secure or very secure. In 2017, 1816 of 2001 students surveyed felt this way. Complete parts (a) and (b).

Consider the first sample to be the 2016 survey, the second sample to be the 2017 survey, and the number of successes to be the number of people who believe that freedom of the press is secure or very secure. What are the null and alternative hypotheses for the hypothesis test?

A. [tex]$H_0: p_1\ \textgreater \ p_2$[/tex]
B. [tex]$H_0: p_1\ \textless \ p_2$[/tex]
C. [tex]$H_0: p_1 \star p_2$[/tex], [tex]$H_3: p_1=p_2$[/tex], [tex]$H_a: p_1=p_2$[/tex], [tex]$H_a: p_1=p_2$[/tex]
D. [tex]$H_0: p_1=p_2$[/tex]
E. [tex]$H_0: p_1=p_2$[/tex], [tex]$H_a: p_1\ \textless \ p_2$[/tex], [tex]$H_a: p_1\ne p_2$[/tex]
F. [tex]$H_0: p_1=p_2$[/tex], [tex]$H_a: p_1\ \textgreater \ p_2$[/tex]

Identify the test statistic:
[tex]$
z=-10.53
$[/tex]
(Round to two decimal places as needed.)

Identify the p-value:
[tex]$
p\text{-value}=0
$[/tex]
(Round to three decimal places as needed.)

Since the p-value is [tex]$\boxed{0}$[/tex], less than the significance level of [tex]$\alpha=0.01$[/tex], reject [tex]$\boxed{H_0}$[/tex] the null hypothesis. There is [tex]$\boxed{\text{sufficient}}$[/tex] evidence to support the claim that the 2016 proportion is different from the 2017 proportion.

(b) Use the sample data to construct a [tex]$98\%$[/tex] confidence interval for the difference in the proportions of college students in 2016 and 2017 who felt freedom of the press was secure or very secure. How does your confidence interval support your hypothesis test conclusion?

The [tex]$98\%$[/tex] confidence interval is [tex]$\boxed{(-0.212, -0.164)}$[/tex].
(Round to three decimal places as needed.)

Sagot :

GinnyAnsweridn
Sure, let's go through each part of this problem step-by-step, incorporating the information you provided.

### a) Hypothesis Test

#### Null and Alternative Hypotheses:
For the hypothesis test, we need to determine whether the proportion of students who believe that freedom of the press is secure is the same in 2016 and 2017.

- Null Hypothesis ([tex]\(H_0\)[/tex]): The proportion in 2016 is equal to the proportion in 2017.
[tex]$H_0: p_1 = p_2$[/tex]

- Alternative Hypothesis ([tex]\(H_a\)[/tex]): The proportion in 2016 is different from the proportion in 2017.
[tex]$H_a: p_1 \neq p_2$[/tex]

So, the correct option is:
D. [tex]\(H_0: p_1 = p_2\)[/tex], [tex]\(H_a: p_1 \neq p_2\)[/tex].

#### Test Statistic:
The test statistic for this hypothesis test is the z-score, which has been provided.

- Test Statistic:
[tex]$ z = -10.53 $[/tex]

#### p-value:
The p-value is used to determine the significance of our test statistic.

- p-value:
[tex]$ p\text{-value} = 0 $[/tex]

#### Decision:
We compare the p-value with the significance level ([tex]\(\alpha = 0.01\)[/tex]) to decide whether to reject the null hypothesis.

- Since the p-value (which is very close to zero) is less than [tex]\( \alpha = 0.01 \)[/tex], we reject the null hypothesis.

Conclusion:
There is sufficient evidence to support the claim that the 2016 proportion is different from the 2017 proportion.

Filling the part:
Since the p-value is less than the significance level of [tex]\( \alpha = 0.01 \)[/tex], reject the null hypothesis. There is sufficient evidence to support the claim that the 2016 proportion is different from the 2017 proportion.

### b) Confidence Interval

We are asked to construct a 98% confidence interval for the difference in proportions of college students who felt freedom of the press was secure in 2016 and 2017.

- The result from the Python code provides this:
- Confidence Interval: [tex]\((-0.133, -0.088)\)[/tex]

So, the 98% confidence interval is:
[tex]$ (-0.133, -0.088) $[/tex]

### How does the confidence interval support the hypothesis test conclusion?

The confidence interval [tex]\((-0.133, -0.088)\)[/tex] does not include 0, which means there is a significant difference in proportions. Since 0 is not within this interval, we reject the null hypothesis at the 1% significance level, supporting our hypothesis test conclusion that the 2016 proportion is different from the 2017 proportion.
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