According to a recent survey of 1410 voters, $$ 72 \% $$ feel that the president is doing an acceptable job. Construct a $$ 93 \% $$ confidence interval to estimate the proportion of voters who feel the president is doing a acceptable job. $$ \widehat{p}= $$ $$ \frac{\alpha}{2}= $$ $$ z_{\frac{\alpha}{2}}= $$ Margin of Error: $$ E= $$ Wetween (round to $$ 43 \% $$ decimal places) benfident that the proportion of voters who feel the president is doing an acceptable job is

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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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