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
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.

(Round up to the nearest integer)
b. Assume that of all adults gamble online.

(Round up to the nearest integer)

Ask by Bush Stuart. in the United States

Mar 20,2025

Question

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
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.

(Round up to the nearest integer)
b. Assume that of all adults gamble online.

(Round up to the nearest integer)

Ask by Bush Stuart. in the United States

Mar 20,2025

UpStudy AI · Accepted Answer

Given a confidence level of $$90\%$$, the corresponding critical value is approximately $$z = 1.645$$. The margin of error required is $$E = 0.02$$ (since two percentage points is 0.02).

We use the sample size formula for a proportion:
$$
n = \frac{z^2 \, p(1-p)}{E^2}
$$

### Part (a)
Assume nothing is known about the percentage of adults who gamble online. In this case, we use $$ p = 0.5 $$ to maximize the product $$ p(1-p) $$.

Substitute the values:
$$
n = \frac{(1.645)^2 \, (0.5)(0.5)}{(0.02)^2}
$$
Since $$(0.5)(0.5) = 0.25$$, we have:
$$
n = \frac{(1.645)^2 \cdot 0.25}{0.0004}
$$
Calculate $$ (1.645)^2 $$:
$$
(1.645)^2 \approx 2.706025
$$
Thus:
$$
n = \frac{2.706025 \cdot 0.25}{0.0004} = \frac{0.67650625}{0.0004} \approx 1691.265625
$$
Since we round up to the nearest integer:
$$
n = 1692
$$

### Part (b)
Assume that $$18\%$$ of all adults gamble online, so $$ p = 0.18 $$ (and $$1 - p = 0.82$$).

Substitute the values:
$$
n = \frac{(1.645)^2 \, (0.18)(0.82)}{(0.02)^2}
$$
Calculate the product $$0.18 \times 0.82$$:
$$
0.18 \times 0.82 = 0.1476
$$
Then, substitute:
$$
n = \frac{2.706025 \cdot 0.1476}{0.0004} \approx \frac{0.399}{0.0004} \approx 997.5
$$
Rounding up to the nearest integer gives:
$$
n = 998
$$
To determine the number of adults to survey: **Part (a):** Assuming no prior knowledge, survey **1,692** adults. **Part (b):** Assuming that 18% gamble online, survey **998** adults.

Upstudy AI Solution

Answer

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Part (a)

Part (b)

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