A little more practice with confidence intervals…
A January 2019 Rasmussen Reports poll asked a random sample of 1000 likely U.S. voters if they believe social media such
Facebook and Twitter make the U.S. a more or less divided nation. Of these likely voters, 650 said they believe social media
makes the U.S. a more divided nation.
Find a 95% confidence interval for the proportion of all likely U.S. voters who believe social media such as Facebook and
Twitter make the U.S. a more divided nation.
Report your answer as an interval in the form (lower bound, upper bound) rounded to three decimal places.

Question

UpStudy AI · Accepted Answer

To calculate a 95% confidence interval for the proportion of all likely U.S. voters who believe social media makes the U.S. a more divided nation, we can use the formula for the confidence interval for a proportion:

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

Where:
- $$\hat{p}$$ is the sample proportion,
- $$z$$ is the z-score corresponding to the desired confidence level,
- $$n$$ is the sample size.

### Step 1: Calculate the sample proportion $$\hat{p}$$

Given:
- Number of voters who believe social media makes the U.S. more divided = 650
- Total number of voters surveyed = 1000

$$
\hat{p} = \frac{650}{1000} = 0.65
$$

### Step 2: Find the z-score for a 95% confidence level

For a 95% confidence level, the z-score is approximately 1.96.

### Step 3: Calculate the standard error (SE)

$$
SE = \sqrt{\frac{\hat{p}(1 - \hat{p})}{n}} = \sqrt{\frac{0.65(1 - 0.65)}{1000}} = \sqrt{\frac{0.65 \times 0.35}{1000}} = \sqrt{\frac{0.2275}{1000}} \approx \sqrt{0.0002275} \approx 0.0151
$$

### Step 4: Calculate the margin of error (ME)

$$
ME = z \cdot SE = 1.96 \cdot 0.0151 \approx 0.0296
$$

### Step 5: Calculate the confidence interval

$$
\text{Lower bound} = \hat{p} - ME = 0.65 - 0.0296 \approx 0.6204
$$
$$
\text{Upper bound} = \hat{p} + ME = 0.65 + 0.0296 \approx 0.6796
$$

### Step 6: Round to three decimal places

The 95% confidence interval for the proportion of all likely U.S. voters who believe social media makes the U.S. a more divided nation is:

$$
(0.620, 0.680)
$$

Thus, the final answer is:

$$
\boxed{(0.620, 0.680)}
$$
The 95% confidence interval is (0.620, 0.680).

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