It's believed that as many as $$ 21 \% $$ of adults over 50 never graduated from high school. We wish to see if this percentage is the same among the 25 to 30 age group. a) How many of this younger age group must we survey in order to estimate the proportion of non-grads to within $$ 8 \% $$ with $$ 90 \% $$ confidence? $$ n=71 $$ (Round up to the nearest integer.) b) Suppose we want to cut the margin of error to $$ 3 \% $$. What is the necessary sample size? $$ n=\square $$ (Round up to the nearest integer.)

Question

UpStudy AI · Accepted Answer

**a) Finding the sample size for an $$8\%$$ margin of error**

We use the formula for the required sample size for a proportion:
$$
n = \left( \frac{z_{\alpha/2}\sqrt{p(1-p)}}{E} \right)^2
$$
where  
$$ p = 0.21 $$ (assumed proportion),  
$$ E = 0.08 $$ (margin of error), and  
for $$90\%$$ confidence, $$ z_{\alpha/2} = 1.645 $$.

1. Compute the standard deviation component:
   $$
   \sqrt{p(1-p)} = \sqrt{0.21 \times 0.79} \approx \sqrt{0.1659} \approx 0.4074.
   $$
2. Multiply by $$ z_{\alpha/2} $$:
   $$
   1.645 \times 0.4074 \approx 0.6703.
   $$
3. Divide by $$ E $$:
   $$
   \frac{0.6703}{0.08} \approx 8.379.
   $$
4. Square the result:
   $$
   (8.379)^2 \approx 70.24.
   $$
5. Rounding up, the required sample size is:
   $$
   n = 71.
   $$

---

**b) Finding the sample size for a $$3\%$$ margin of error**

We use the same formula:
$$
n = \left( \frac{z_{\alpha/2}\sqrt{p(1-p)}}{E} \right)^2,
$$
with  
$$ E = 0.03 $$.

1. The numerator remains the same:
   $$
   1.645 \times 0.4074 \approx 0.6703.
   $$
2. Divide by $$ E $$:
   $$
   \frac{0.6703}{0.03} \approx 22.343.
   $$
3. Square the result:
   $$
   (22.343)^2 \approx 499.14.
   $$
4. Rounding up, the required sample size is:
   $$
   n = 500.
   $$

---

Final Answers:

a) $$ n = 71 $$  
b) $$ n = 500 $$
a) To estimate the proportion of non-graduates within 8% with 90% confidence, you need to survey **71** people. b) To reduce the margin of error to 3%, you need to survey **500** people.

Upstudy AI Solution

Answer

Solution

Beyond the Answer

a) Calculating for 8% Margin of Error

b) Calculating for 3% Margin of Error

Related Questions

Latest Statistics Questions