Answer :
Let's break down the given question and solution step-by-step for parts (a) and (b).
### Part (a)
Determine whether the proportion of college students who believe that freedom of the press is secure or very secure in this country has changed from 2016 to 2017. Use a significance level of 0.01.
First, let's set up our null and alternative hypotheses.
#### Hypotheses:
- Null Hypothesis (H₀): The proportion of college students who believe that freedom of the press is secure or very secure is the same in 2016 and 2017.
- Alternative Hypothesis (Hₐ): The proportion of college students who believe that freedom of the press is secure or very secure is different in 2016 and 2017.
Thus, the correct choice is:
[tex]\[ H_0: P_1 = P_2 \][/tex]
[tex]\[ H_a: P_1 \ne P_2 \][/tex]
This corresponds to option D.
#### Test Statistic:
The calculated test statistic (z) is the value used to determine the p-value and make a conclusion about the null hypothesis.
From the solution, the test statistic is [tex]\( z = -71.58 \)[/tex].
#### P-value:
The p-value helps to determine the significance of the results. It's essentially the probability of obtaining test results at least as extreme as the results actually observed, under the assumption that the null hypothesis is correct.
From the solution, the p-value is [tex]\( p\_value = 0 \)[/tex].
#### Decision Rule:
Since the significance level ([tex]\( \alpha \)[/tex]) is 0.01, we compare this to our p-value:
- If [tex]\( p\_value < \alpha \)[/tex], we reject the null hypothesis.
- If [tex]\( p\_value \ge \alpha \)[/tex], we do not reject the null hypothesis.
In this case, since [tex]\( p\_value = 0 \)[/tex] which is less than [tex]\( \alpha = 0.01 \)[/tex], we reject the null hypothesis.
Thus, we conclude there is sufficient evidence to support the claim that the 2016 proportion is different from the 2017 proportion.
### Part (b)
Use the sample data to construct a 95% 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?
#### Confidence Interval:
To construct a 95% confidence interval for the difference in proportions, the solution provides the interval as:
[tex]\[ (-1.607, -1.607) \][/tex]
This interval indicates that the true difference in proportions lies somewhere between these values.
Since the entire confidence interval is negative and does not include zero, this supports our hypothesis test conclusion. Specifically, it suggests that there is a significant difference between the proportions for the years 2016 and 2017, making it reasonable to believe that the proportion has indeed changed between these two years.
### Summary of Results:
1. Null Hypothesis: [tex]\( H_0: P_1 = P_2 \)[/tex]
2. Alternative Hypothesis: [tex]\( H_a: P_1 \ne P_2 \)[/tex]
3. Test Statistic (z): [tex]\( -71.58 \)[/tex]
4. P-value: [tex]\( 0 \)[/tex]
5. Decision: Reject the null hypothesis (since p-value [tex]\( < \)[/tex] significance level [tex]\( \alpha = 0.01 \)[/tex]).
6. Conclusion: There is sufficient evidence to support the claim that the 2016 proportion is different from the 2017 proportion.
7. 95% Confidence Interval: [tex]\((-1.607, -1.607)\)[/tex], which supports the conclusion that the proportions are significantly different.
### Part (a)
Determine whether the proportion of college students who believe that freedom of the press is secure or very secure in this country has changed from 2016 to 2017. Use a significance level of 0.01.
First, let's set up our null and alternative hypotheses.
#### Hypotheses:
- Null Hypothesis (H₀): The proportion of college students who believe that freedom of the press is secure or very secure is the same in 2016 and 2017.
- Alternative Hypothesis (Hₐ): The proportion of college students who believe that freedom of the press is secure or very secure is different in 2016 and 2017.
Thus, the correct choice is:
[tex]\[ H_0: P_1 = P_2 \][/tex]
[tex]\[ H_a: P_1 \ne P_2 \][/tex]
This corresponds to option D.
#### Test Statistic:
The calculated test statistic (z) is the value used to determine the p-value and make a conclusion about the null hypothesis.
From the solution, the test statistic is [tex]\( z = -71.58 \)[/tex].
#### P-value:
The p-value helps to determine the significance of the results. It's essentially the probability of obtaining test results at least as extreme as the results actually observed, under the assumption that the null hypothesis is correct.
From the solution, the p-value is [tex]\( p\_value = 0 \)[/tex].
#### Decision Rule:
Since the significance level ([tex]\( \alpha \)[/tex]) is 0.01, we compare this to our p-value:
- If [tex]\( p\_value < \alpha \)[/tex], we reject the null hypothesis.
- If [tex]\( p\_value \ge \alpha \)[/tex], we do not reject the null hypothesis.
In this case, since [tex]\( p\_value = 0 \)[/tex] which is less than [tex]\( \alpha = 0.01 \)[/tex], we reject the null hypothesis.
Thus, we conclude there is sufficient evidence to support the claim that the 2016 proportion is different from the 2017 proportion.
### Part (b)
Use the sample data to construct a 95% 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?
#### Confidence Interval:
To construct a 95% confidence interval for the difference in proportions, the solution provides the interval as:
[tex]\[ (-1.607, -1.607) \][/tex]
This interval indicates that the true difference in proportions lies somewhere between these values.
Since the entire confidence interval is negative and does not include zero, this supports our hypothesis test conclusion. Specifically, it suggests that there is a significant difference between the proportions for the years 2016 and 2017, making it reasonable to believe that the proportion has indeed changed between these two years.
### Summary of Results:
1. Null Hypothesis: [tex]\( H_0: P_1 = P_2 \)[/tex]
2. Alternative Hypothesis: [tex]\( H_a: P_1 \ne P_2 \)[/tex]
3. Test Statistic (z): [tex]\( -71.58 \)[/tex]
4. P-value: [tex]\( 0 \)[/tex]
5. Decision: Reject the null hypothesis (since p-value [tex]\( < \)[/tex] significance level [tex]\( \alpha = 0.01 \)[/tex]).
6. Conclusion: There is sufficient evidence to support the claim that the 2016 proportion is different from the 2017 proportion.
7. 95% Confidence Interval: [tex]\((-1.607, -1.607)\)[/tex], which supports the conclusion that the proportions are significantly different.
