A computer manufacturing company has sent a mail survey to 2,800 of its randomly selected customers that have purchased a new laptop. The survey asked the customers whether or not they were satisfied with the computer. Eight hundred custpmers responded to the survey: 640 customers indicated that they were satisfied, while 160 customers indicated they were not satisfied with their new computer. Construct a 96 percent confidence interval estimate of the true proportion of customers satisfied with their new computer. $$ [.788, .812] $$

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To construct a 96 percent confidence interval for the true proportion of customers satisfied with their new computer, we can follow these steps:

### Step 1: Identify the known values
- Total number of respondents, $$ n = 800 $$
- Number of satisfied customers, $$ x = 640 $$

### Step 2: Calculate the sample proportion
The sample proportion $$ \hat{p} $$ is calculated as follows:
$$
\hat{p} = \frac{x}{n} = \frac{640}{800}
$$

### Step 3: Calculate the standard error
The standard error (SE) of the sample proportion is given by the formula:
$$
SE = \sqrt{\frac{\hat{p}(1 - \hat{p})}{n}}
$$

### Step 4: Find the z-score for the confidence level
For a 96% confidence level, we need to find the z-score that corresponds to the critical value. The z-score for a 96% confidence level can be found using a z-table or calculator. The critical value $$ z $$ for 96% confidence is approximately $$ 2.05 $$.

### Step 5: Calculate the margin of error
The margin of error (ME) is calculated as:
$$
ME = z \times SE
$$

### Step 6: Construct the confidence interval
The confidence interval is given by:
$$
\left( \hat{p} - ME, \hat{p} + ME \right)
$$

Now, let's perform the calculations step by step.

1. Calculate $$ \hat{p} $$.
2. Calculate $$ SE $$.
3. Calculate $$ ME $$.
4. Construct the confidence interval.

Let's perform these calculations.
Calculate the value by following steps:
- step0: Calculate:
  $$\frac{640}{800}$$
- step1: Reduce the fraction:
  $$\frac{4}{5}$$
Calculate or simplify the expression $$ \sqrt((640/800)*(1-(640/800))/800) $$.
Calculate the value by following steps:
- step0: Calculate:
  $$\sqrt{\frac{\left(\frac{640}{800}\right)\left(1-\left(\frac{640}{800}\right)\right)}{800}}$$
- step1: Reduce the fraction:
  $$\sqrt{\frac{\frac{4}{5}\left(1-\left(\frac{640}{800}\right)\right)}{800}}$$
- step2: Reduce the fraction:
  $$\sqrt{\frac{\frac{4}{5}\left(1-\frac{4}{5}\right)}{800}}$$
- step3: Subtract the numbers:
  $$\sqrt{\frac{\frac{4}{5}\times \frac{1}{5}}{800}}$$
- step4: Multiply the numbers:
  $$\sqrt{\frac{\frac{4}{25}}{800}}$$
- step5: Divide the terms:
  $$\sqrt{\frac{1}{5000}}$$
- step6: Use the properties of radicals:
  $$\frac{\sqrt{1}}{\sqrt{5000}}$$
- step7: Simplify the expression:
  $$\frac{1}{50\sqrt{2}}$$
- step8: Multiply by the Conjugate:
  $$\frac{\sqrt{2}}{50\sqrt{2}\times \sqrt{2}}$$
- step9: Multiply the numbers:
  $$\frac{\sqrt{2}}{100}$$
Calculate or simplify the expression $$ 2.05 * 0.014142 $$.
Calculate the value by following steps:
- step0: Calculate:
  $$2.05\times 0.014142$$
- step1: Multiply the numbers:
  $$0.0289911$$
Calculate or simplify the expression $$ 0.8 - 0.0289911 $$.
Calculate the value by following steps:
- step0: Calculate:
  $$0.8-0.0289911$$
- step1: Subtract the numbers:
  $$0.7710089$$
Calculate or simplify the expression $$ 0.8 + 0.0289911 $$.
Calculate the value by following steps:
- step0: Calculate:
  $$0.8+0.0289911$$
- step1: Add the numbers:
  $$0.8289911$$
### Step-by-Step Calculations

1. **Calculate the sample proportion $$ \hat{p} $$**:
   $$
   \hat{p} = \frac{640}{800} = 0.8
   $$

2. **Calculate the standard error (SE)**:
   $$
   SE = \sqrt{\frac{0.8(1 - 0.8)}{800}} = \sqrt{\frac{0.8 \times 0.2}{800}} \approx 0.014142
   $$

3. **Find the margin of error (ME)**:
   $$
   ME = z \times SE = 2.05 \times 0.014142 \approx 0.0289911
   $$

4. **Construct the confidence interval**:
   - Lower limit:
   $$
   \hat{p} - ME = 0.8 - 0.0289911 \approx 0.7710089
   $$
   - Upper limit:
   $$
   \hat{p} + ME = 0.8 + 0.0289911 \approx 0.8289911
   $$

### Final Confidence Interval
Thus, the 96% confidence interval estimate of the true proportion of customers satisfied with their new computer is approximately:
$$
[0.771, 0.829]
$$

This result indicates that we can be 96% confident that the true proportion of satisfied customers lies within this interval.
The 96% confidence interval for the true proportion of satisfied customers is [0.771, 0.829].

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