The population proportion is assumed to be 0.45 . If the desired standard error is 0.0174 or less, what is the minimum sample size needed?

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To find the minimum sample size needed for a given standard error, we can use the formula for the standard error of a proportion:

$$ \text{Standard Error} = \sqrt{\frac{p(1-p)}{n}} $$

where:
- $$ p $$ is the population proportion,
- $$ n $$ is the sample size.

Given:
- Population proportion ($$ p $$) = 0.45
- Desired standard error = 0.0174 or less

We need to find the minimum sample size ($$ n $$) that will result in a standard error of 0.0174 or less.

Let's rearrange the formula to solve for $$ n $$:

$$ n = \frac{p(1-p)}{\text{Standard Error}^2} $$

Substitute the given values into the formula:

$$ n = \frac{0.45(1-0.45)}{0.0174^2} $$

Now, we can calculate the minimum sample size needed.
Calculate the value by following steps:
- step0: Calculate:
  $$\frac{0.45\left(1-0.45\right)}{0.0174^{2}}$$
- step1: Subtract the numbers:
  $$\frac{0.45\times 0.55}{0.0174^{2}}$$
- step2: Convert the expressions:
  $$\frac{0.45\times 0.55}{\left(\frac{87}{5000}\right)^{2}}$$
- step3: Multiply the numbers:
  $$\frac{0.2475}{\left(\frac{87}{5000}\right)^{2}}$$
- step4: Convert the expressions:
  $$\frac{\frac{99}{400}}{\left(\frac{87}{5000}\right)^{2}}$$
- step5: Evaluate the power:
  $$\frac{\frac{99}{400}}{\frac{7569}{5000^{2}}}$$
- step6: Multiply by the reciprocal:
  $$\frac{99}{400}\times \frac{5000^{2}}{7569}$$
- step7: Reduce the numbers:
  $$\frac{11}{400}\times \frac{5000^{2}}{841}$$
- step8: Rewrite the expression:
  $$\frac{11}{400}\times \frac{200^{2}\times 25^{2}}{841}$$
- step9: Rewrite the expression:
  $$\frac{11}{200\times 2}\times \frac{200^{2}\times 25^{2}}{841}$$
- step10: Reduce the numbers:
  $$11\times \frac{100\times 25^{2}}{841}$$
- step11: Multiply:
  $$\frac{11\times 62500}{841}$$
- step12: Multiply:
  $$\frac{687500}{841}$$
The minimum sample size needed to achieve a standard error of 0.0174 or less is approximately 817.48.
The minimum sample size needed is approximately 818.

The population proportion is assumed to be 0.45 . If the desired standard error is 0.0174 or less, what is the minimum sample size needed?

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