You work for a marketing firm that has a large client in the automobile industry, You have been asked to estimate the proportion of households in chicago that have two or more vehicles. You have been assigned to gather a random sample that could be used to estimate this proportion to within 3 percentage points of the true population of households in Chicago that have two or more vehicles with a $$ 80 \% $$ level of confidence, a) With no prior research, what sample size should you gather to make such an estimation? Round your answer up to the nearest whole number, $$ n \approx $$ Enter your answer $$ \begin{array}{ll} ext { b) Your firm has decided that your plan is too expensive, and they wish to reduce the sample size required, You conduct a small preliminary sample, and you obtain a sample } \ ext { proportion of } p^{\prime}=0.19 ext {. Using this new information, what sample size should you gather in order to obtain the estimation as specified in the question above? Round your } \ ext { answer up to the nearest whole number. } \ n \approx ext { Enter your answer } & ext { vaseholds households }\end{array} $$

Question

You work for a marketing firm that has a large client in the automobile industry, You have been asked to estimate the proportion of households in chicago that have two or more vehicles. You have been assigned to gather a random sample that could be used to estimate this proportion to within 3 percentage points of the true population of households in Chicago that have two or more vehicles with a $$ 80 \% $$ level of confidence, a) With no prior research, what sample size should you gather to make such an estimation? Round your answer up to the nearest whole number, $$ n \approx $$ Enter your answer $$ \begin{array}{ll}	ext { b) Your firm has decided that your plan is too expensive, and they wish to reduce the sample size required, You conduct a small preliminary sample, and you obtain a sample } \ 	ext { proportion of } p^{\prime}=0.19 	ext {. Using this new information, what sample size should you gather in order to obtain the estimation as specified in the question above? Round your } \ 	ext { answer up to the nearest whole number. } \ n \approx 	ext { Enter your answer } & 	ext { vaseholds households }\end{array} $$

UpStudy AI · Accepted Answer

Calculating $$ n $$ using the formula with $$ Z = 1.28 $$, $$ p = 0.5 $$, and $$ E = 0.03 $$.
For part b, we will use the sample proportion $$ p' = 0.19 $$ to calculate the new sample size.1. **Identify the parameters:**
   - $$ p' = 0.19 $$
   - $$ E = 0.03 $$

2. **Calculate the sample size:**
   $$
   n = \left( \frac{(1.28)^2 \cdot 0.19(1-0.19)}{(0.03)^2} \right)
   $$

Now, let's calculate this value using a tool call.
Calculating $$ n $$ using the formula with $$ Z = 1.28 $$, $$ p' = 0.19 $$, and $$ E = 0.03 $$.Now, I will perform the calculations for both parts.

### Part a Calculation
Calculating $$ n $$ for part a.functions.calculate_or_simplify  { "expression": "(1.28^2 * 0.5 * (1 - 0.5)) / (0.03^2)", "return_form": "number" }

### Part b Calculation
Calculating $$ n $$ for part b.functions.calculate_or_simplify  { "expression": "(1.28^2 * 0.19 * (1 - 0.19)) / (0.03^2)", "return_form": "number" }

Now, let's proceed with the calculations.
Calculate the value by following steps:
- step0: Calculate:
  $$\frac{\left(1.28^{2}\times 0.5\left(1-0.5\right)\right)}{0.03^{2}}$$
- step1: Remove the parentheses:
  $$\frac{1.28^{2}\times 0.5\left(1-0.5\right)}{0.03^{2}}$$
- step2: Subtract the numbers:
  $$\frac{1.28^{2}\times 0.5\times 0.5}{0.03^{2}}$$
- step3: Convert the expressions:
  $$\frac{\left(\frac{32}{25}\right)^{2}\times 0.5\times 0.5}{0.03^{2}}$$
- step4: Convert the expressions:
  $$\frac{\left(\frac{32}{25}\right)^{2}\times 0.5\times 0.5}{\left(\frac{3}{100}\right)^{2}}$$
- step5: Multiply:
  $$\frac{\frac{2^{8}}{25^{2}}}{\left(\frac{3}{100}\right)^{2}}$$
- step6: Evaluate the power:
  $$\frac{\frac{2^{8}}{25^{2}}}{\frac{9}{10000}}$$
- step7: Multiply by the reciprocal:
  $$\frac{2^{8}}{25^{2}}\times \frac{10000}{9}$$
- step8: Rewrite the expression:
  $$\frac{2^{8}}{25^{2}}\times \frac{625\times 16}{9}$$
- step9: Rewrite the expression:
  $$\frac{2^{8}}{25^{2}}\times \frac{25^{2}\times 16}{9}$$
- step10: Reduce the numbers:
  $$2^{8}\times \frac{16}{9}$$
- step11: Multiply:
  $$\frac{2^{8}\times 16}{9}$$
- step12: Multiply:
  $$\frac{2^{12}}{9}$$
- step13: Evaluate the power:
  $$\frac{4096}{9}$$
Calculate or simplify the expression $$ (1.28^2 * 0.19 * (1 - 0.19)) / (0.03^2) $$.
Calculate the value by following steps:
- step0: Calculate:
  $$\frac{\left(1.28^{2}\times 0.19\left(1-0.19\right)\right)}{0.03^{2}}$$
- step1: Remove the parentheses:
  $$\frac{1.28^{2}\times 0.19\left(1-0.19\right)}{0.03^{2}}$$
- step2: Subtract the numbers:
  $$\frac{1.28^{2}\times 0.19\times 0.81}{0.03^{2}}$$
- step3: Convert the expressions:
  $$\frac{\left(\frac{32}{25}\right)^{2}\times 0.19\times 0.81}{0.03^{2}}$$
- step4: Convert the expressions:
  $$\frac{\left(\frac{32}{25}\right)^{2}\times 0.19\times 0.81}{\left(\frac{3}{100}\right)^{2}}$$
- step5: Multiply:
  $$\frac{\frac{98496}{25^{4}}}{\left(\frac{3}{100}\right)^{2}}$$
- step6: Evaluate the power:
  $$\frac{\frac{98496}{25^{4}}}{\frac{9}{10000}}$$
- step7: Multiply by the reciprocal:
  $$\frac{98496}{25^{4}}\times \frac{10000}{9}$$
- step8: Reduce the numbers:
  $$\frac{10944}{25^{4}}\times 10000$$
- step9: Rewrite the expression:
  $$\frac{10944}{25^{4}}\times 625\times 16$$
- step10: Rewrite the expression:
  $$\frac{10944}{25^{4}}\times 25^{2}\times 16$$
- step11: Reduce the numbers:
  $$\frac{10944}{25^{2}}\times 16$$
- step12: Multiply:
  $$\frac{10944\times 16}{25^{2}}$$
- step13: Multiply:
  $$\frac{175104}{25^{2}}$$
- step14: Evaluate the power:
  $$\frac{175104}{625}$$
The calculated sample size for part a is approximately $$ 455.11 $$. Rounding up gives $$ n \approx 456 $$.
The calculated sample size for part b is approximately $$ 280.1664 $$. Rounding up gives $$ n \approx 281 $$.
For part a, $$ n \approx 456 $$. For part b, $$ n \approx 281 $$.

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