Questions: A poll asked college students in 2016 and again in 2017 whether they believed the First Amendment guarantee of freedom of religion was secure or threatened in the country today. In 2016, 2064 of 3091 students surveyed said that freedom of religion was secure or very secure. In 2017, 1953 of 2987 students surveyed felt this way. Complete parts (a) and (b). Identify the test statistic. z= (Round to two decimal places as needed.)

A poll asked college students in 2016 and again in 2017 whether they believed the First Amendment guarantee of freedom of religion was secure or threatened in the country today. In 2016, 2064 of 3091 students surveyed said that freedom of religion was secure or very secure. In 2017, 1953 of 2987 students surveyed felt this way. Complete parts (a) and (b).
Identify the test statistic.
z=
(Round to two decimal places as needed.)

Transcript text: A poll asked college students in 2016 and again in 2017 whether they believed the First Amendment guarantee of freedom of religion was secure of threatened in the country today. In 2016, 2064 of 3091 students surveyed said that freedom of religion was secure or very secure. In 2017,1953 of 2987 students surveyed felt this way. Complete parts (a) and (b). Identify the test statistic. $\mathrm{z}=$ $\square$ (Round to two decimal places as needed.)

Solution

Solution Steps

Step 1: Define the Problem

We are tasked with determining whether the belief in the security of the First Amendment guarantee of freedom of religion among college students has changed from 2016 to 2017. We will calculate the z-statistic to compare the proportions of students who felt that freedom of religion was secure in both years.

Step 2: Gather Data

From the survey results:

In 2016, $ n_1 = 3091 $ students were surveyed, and $ X_1 = 2064 $ believed that freedom of religion was secure.
In 2017, $ n_2 = 2987 $ students were surveyed, and $ X_2 = 1953 $ believed that freedom of religion was secure.

Step 3: Calculate Sample Proportions

The sample proportions for each year are calculated as follows: \[ p_1 = \frac{X_1}{n_1} = \frac{2064}{3091} \approx 0.6667 \] \[ p_2 = \frac{X_2}{n_2} = \frac{1953}{2987} \approx 0.6545 \]

Step 4: Calculate Pooled Sample Proportion

The pooled sample proportion is given by: \[ p_{\text{pooled}} = \frac{X_1 + X_2}{n_1 + n_2} = \frac{2064 + 1953}{3091 + 2987} \approx 0.6610 \]

Step 5: Calculate Standard Error

The standard error (SE) for the difference in proportions is calculated as: \[ SE = \sqrt{p_{\text{pooled}} \cdot (1 - p_{\text{pooled}}) \left( \frac{1}{n_1} + \frac{1}{n_2} \right)} \approx 0.0121 \]

Step 6: Calculate the Z-Statistic

The z-statistic is calculated using the formula: \[ z = \frac{p_1 - p_2}{SE} = \frac{0.6667 - 0.6545}{0.0121} \approx 1.15 \]