Use the given data to find the minimum sample size required to estimato the population proportion.
Margin of error 0.008 , confidence lever ; p and qunknown
OA. 1116
O B. 2223
O C. 1929
OD. 2115

Question

Use the given data to find the minimum sample size required to estimato the population proportion.
Margin of error 0.008 , confidence lever

; p and qunknown
OA. 1116
O B. 2223
O C. 1929
OD. 2115

UpStudy AI · Accepted Answer

To find the minimum sample size required to estimate the population proportion with a given margin of error and confidence level, we can use the formula:

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

Where:
- $$ n $$ = required sample size
- $$ Z $$ = Z-value corresponding to the confidence level
- $$ p $$ = estimated proportion (since $$ p $$ is unknown, we can use $$ 0.5 $$ for maximum variability)
- $$ q = 1 - p $$
- $$ E $$ = margin of error

Given:
- Margin of error $$ E = 0.008 $$
- Confidence level $$ 99\% $$

First, we need to find the Z-value for a $$ 99\% $$ confidence level. The Z-value for $$ 99\% $$ confidence is approximately $$ 2.576 $$.

Now, we can calculate $$ p $$ and $$ q $$:
- $$ p = 0.5 $$
- $$ q = 1 - p = 0.5 $$

Now we can substitute these values into the formula to find $$ n $$:

$$
n = \left( \frac{(2.576)^2 \cdot (0.5) \cdot (0.5)}{(0.008)^2} \right)
$$

Let's calculate this step by step. First, we will compute the components of the formula.
Calculate the value by following steps:
- step0: Calculate:
  $$0.008^{2}$$
- step1: Convert the expressions:
  $$\left(\frac{1}{125}\right)^{2}$$
- step2: Use the properties of exponents:
  $$\frac{1^{2}}{125^{2}}$$
- step3: Evaluate the power:
  $$\frac{1}{125^{2}}$$
- step4: Evaluate the power:
  $$\frac{1}{15625}$$
Calculate or simplify the expression $$ (2.576)^2 $$.
Calculate the value by following steps:
- step0: Calculate:
  $$2.576^{2}$$
- step1: Convert the expressions:
  $$\left(\frac{322}{125}\right)^{2}$$
- step2: Use the properties of exponents:
  $$\frac{322^{2}}{125^{2}}$$
- step3: Evaluate the power:
  $$\frac{322^{2}}{15625}$$
The calculations yield the following results:

1. $$ (2.576)^2 \approx 6.635776 $$
2. $$ (0.008)^2 = 0.000064 $$

Now we can substitute these values back into the sample size formula:

$$
n = \left( \frac{6.635776 \cdot 0.5 \cdot 0.5}{0.000064} \right)
$$

This simplifies to:

$$
n = \left( \frac{6.635776 \cdot 0.25}{0.000064} \right)
$$

Now, let's calculate this expression.
Calculate the value by following steps:
- step0: Calculate:
  $$\frac{\left(6.635776\times 0.25\right)}{\frac{1}{15625}}$$
- step1: Remove the parentheses:
  $$\frac{6.635776\times 0.25}{\frac{1}{15625}}$$
- step2: Multiply the numbers:
  $$\frac{1.658944}{\frac{1}{15625}}$$
- step3: Convert the expressions:
  $$\frac{\frac{25921}{15625}}{\frac{1}{15625}}$$
- step4: Multiply by the reciprocal:
  $$\frac{25921}{15625}\times 15625$$
- step5: Reduce the numbers:
  $$25921\times 1$$
- step6: Simplify:
  $$25921$$
The calculated sample size $$ n $$ is approximately $$ 25921 $$.

However, this value seems to be inconsistent with the provided options. Let's check if we need to round or adjust based on the context of the problem.

Given the options:
- A. 1116
- B. 2223
- C. 1929
- D. 2115

It appears that the calculated sample size is significantly larger than the options provided. This discrepancy suggests that there may be a misunderstanding in the interpretation of the margin of error or the confidence level.

To ensure accuracy, let's verify the calculations and assumptions made. The formula and values used are correct, but the margin of error might need to be reconsidered or the options might not align with the calculated sample size.

If you have any additional context or constraints, please provide them, and we can reassess the situation.
The minimum sample size required is approximately 25921.

Use the given data to find the minimum sample size required to estimato the population proportion.
Margin of error 0.008 , confidence lever ; p and qunknown
OA. 1116
O B. 2223
O C. 1929
OD. 2115

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Use the given data to find the minimum sample size required to estimato the population proportion. Margin of error 0.008 , confidence lever ; p and qunknown OA. 1116 O B. 2223 O C. 1929 OD. 2115

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Use the given data to find the minimum sample size required to estimato the population proportion.
Margin of error 0.008 , confidence lever ; p and qunknown
OA. 1116
O B. 2223
O C. 1929
OD. 2115