Some states now allow online gambling. As a marketing manager for a casino, you need to determine the percentage of adults in those states who gamble online. How many adults must you survey in order to be $$ 90 \% $$ confident that your estimate is in error by no more than two percentage points? Complete parts (a) and (b) below a. Assume that nothing is known about the percentage of adults who gamble online. $$ \mathrm{n}=\mathrm{T} $$ (Round up to the nearest integer.)

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UpStudy AI · Accepted Answer

To determine the sample size needed for estimating the percentage of adults who gamble online with a specified confidence level and margin of error, we can use the formula for sample size in proportion estimation:

$$
n = \left( \frac{Z^2 \cdot p \cdot (1 - p)}{E^2} \right)
$$

Where:
- $$ n $$ = required sample size
- $$ Z $$ = Z-value corresponding to the desired confidence level
- $$ p $$ = estimated proportion of the population (since nothing is known, we use $$ p = 0.5 $$ for maximum variability)
- $$ E $$ = margin of error

### Given:
- Confidence level = $$ 90\% $$
- Margin of error $$ E = 0.02 $$ (two percentage points)

### Step 1: Find the Z-value
For a $$ 90\% $$ confidence level, the Z-value can be found using a Z-table or standard normal distribution. The Z-value for $$ 90\% $$ confidence is approximately $$ 1.645 $$.

### Step 2: Substitute the values into the formula
Using $$ p = 0.5 $$:

$$
n = \left( \frac{(1.645)^2 \cdot 0.5 \cdot (1 - 0.5)}{(0.02)^2} \right)
$$

Now, let's calculate this expression step by step.
Calculate the value by following steps:
- step0: Calculate:
  $$\frac{1.645^{2}\times 0.5\left(1-0.5\right)}{0.02^{2}}$$
- step1: Subtract the numbers:
  $$\frac{1.645^{2}\times 0.5\times 0.5}{0.02^{2}}$$
- step2: Convert the expressions:
  $$\frac{\left(\frac{329}{200}\right)^{2}\times 0.5\times 0.5}{0.02^{2}}$$
- step3: Convert the expressions:
  $$\frac{\left(\frac{329}{200}\right)^{2}\times 0.5\times 0.5}{\left(\frac{1}{50}\right)^{2}}$$
- step4: Multiply:
  $$\frac{\frac{329^{2}}{160000}}{\left(\frac{1}{50}\right)^{2}}$$
- step5: Evaluate the power:
  $$\frac{\frac{329^{2}}{160000}}{\frac{1}{2500}}$$
- step6: Multiply by the reciprocal:
  $$\frac{329^{2}}{160000}\times 2500$$
- step7: Reduce the numbers:
  $$\frac{329^{2}}{64}\times 1$$
- step8: Multiply:
  $$\frac{329^{2}}{64}$$
Calculate or simplify the expression $$ (1.645)^2 * 0.5 * (1 - 0.5) / (0.02)^2 $$.
Calculate the value by following steps:
- step0: Calculate:
  $$\frac{1.645^{2}\times 0.5\left(1-0.5\right)}{0.02^{2}}$$
- step1: Subtract the numbers:
  $$\frac{1.645^{2}\times 0.5\times 0.5}{0.02^{2}}$$
- step2: Convert the expressions:
  $$\frac{\left(\frac{329}{200}\right)^{2}\times 0.5\times 0.5}{0.02^{2}}$$
- step3: Convert the expressions:
  $$\frac{\left(\frac{329}{200}\right)^{2}\times 0.5\times 0.5}{\left(\frac{1}{50}\right)^{2}}$$
- step4: Multiply:
  $$\frac{\frac{329^{2}}{160000}}{\left(\frac{1}{50}\right)^{2}}$$
- step5: Evaluate the power:
  $$\frac{\frac{329^{2}}{160000}}{\frac{1}{2500}}$$
- step6: Multiply by the reciprocal:
  $$\frac{329^{2}}{160000}\times 2500$$
- step7: Reduce the numbers:
  $$\frac{329^{2}}{64}\times 1$$
- step8: Multiply:
  $$\frac{329^{2}}{64}$$
The calculation yields approximately $$ 1691.265625 $$. Since we need to round up to the nearest integer, we will round this to $$ 1692 $$.

### Final Answer:
a. The number of adults you must survey is $$ n = 1692 $$.
You need to survey 1,692 adults to be 90% confident that your estimate is within 2 percentage points of the true proportion.

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