According to a recent survey of 1410 voters, feel that the president is doing an acceptable job.
Construct a confidence interval to estimate the proportion of voters who feel the president is doing a
acceptable job.

Margin of Error:
Wetween (round to decimal places)
benfident that the proportion of voters who feel the president is doing an acceptable job is

Ask by Lane Hobbs. in the United States

Jan 22,2025

Question

According to a recent survey of 1410 voters, feel that the president is doing an acceptable job.
Construct a confidence interval to estimate the proportion of voters who feel the president is doing a
acceptable job.

Margin of Error:
Wetween (round to decimal places)
benfident that the proportion of voters who feel the president is doing an acceptable job is

Ask by Lane Hobbs. in the United States

Jan 22,2025

UpStudy AI · Accepted Answer

To construct a **93% confidence interval** for the proportion of voters who feel the president is doing an acceptable job, we'll follow these steps:

### 1. **Sample Proportion ($$ \widehat{p} $$)**
Given that 72% of the 1410 voters surveyed feel the president is doing an acceptable job:
$$
\widehat{p} = 0.72
$$

### 2. **Determine $$ \alpha $$ and $$ \frac{\alpha}{2} $$**
The confidence level is 93%, so:
$$
\alpha = 1 - 0.93 = 0.07
$$
$$
\frac{\alpha}{2} = \frac{0.07}{2} = 0.035
$$

### 3. **Find the Critical Value ($$ z_{\frac{\alpha}{2}} $$)**
We need the z-score that corresponds to the upper $$ \frac{\alpha}{2} = 0.035 $$ tail of the standard normal distribution. This is equivalent to finding the z-score for a cumulative probability of $$ 1 - 0.035 = 0.965 $$.

Using standard normal distribution tables or a calculator:
$$
z_{0.035} \approx 1.81
$$

### 4. **Calculate the Margin of Error ($$ E $$)**
The formula for the margin of error for a proportion is:
$$
E = z_{\frac{\alpha}{2}} \times \sqrt{\frac{\widehat{p}(1 - \widehat{p})}{n}}
$$
Plugging in the values:
$$
E = 1.81 \times \sqrt{\frac{0.72 \times 0.28}{1410}} \approx 1.81 \times \sqrt{\frac{0.2016}{1410}} \approx 1.81 \times \sqrt{0.000143} \approx 1.81 \times 0.01196 \approx 0.0217
$$
$$
E \approx 0.0217 \text{ or } 2.17\%
$$

### 5. **Construct the Confidence Interval**
$$
\widehat{p} \pm E = 0.72 \pm 0.0217
$$
$$
\text{Lower bound} = 0.72 - 0.0217 = 0.6983 \quad (69.83\%)
$$
$$
\text{Upper bound} = 0.72 + 0.0217 = 0.7417 \quad (74.17\%)
$$

### **Final 93% Confidence Interval**
$$
\boxed{[69.83\%,\ 74.17\%]}
$$
We are **93% confident** that the true proportion of voters who feel the president is doing an acceptable job lies between **69.83% and 74.17%**.
We are 93% confident that between 69.83% and 74.17% of voters feel the president is doing an acceptable job.

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