For $$ C=90 \% $$ if $$ \sigma=7.21 $$ and $$ E=2 $$ then Sample Size

Question

UpStudy AI · Accepted Answer

1. Identify the $$ z $$-score for a 90% confidence level. For a two-tailed test with confidence level $$ C=90\% $$, the significance level is $$ \alpha=0.10 $$ and $$ \alpha/2=0.05 $$. The corresponding $$ z $$-score is  
   $$
   z_{0.95}=1.645.
   $$

2. Use the sample size formula based on the margin of error $$ E $$, standard deviation $$ \sigma $$, and $$ z $$-score:
   $$
   n = \left( \frac{z_{\frac{\alpha}{2}} \sigma}{E} \right)^2.
   $$

3. Substitute the given values $$ \sigma = 7.21 $$, $$ E = 2 $$, and $$ z_{0.95} = 1.645 $$ into the formula:
   $$
   n = \left( \frac{1.645 \times 7.21}{2} \right)^2.
   $$

4. Compute the numerator:
   $$
   1.645 \times 7.21 \approx 11.86045.
   $$

5. Divide by the margin of error:
   $$
   \frac{11.86045}{2} \approx 5.930225.
   $$

6. Square the result:
   $$
   n \approx (5.930225)^2 \approx 35.1649.
   $$

7. Since the sample size must be a whole number, round up to ensure the margin of error is not exceeded:
   $$
   n = 36.
   $$

Thus, the required sample size is $$ 36 $$.
The required sample size is 36.

For \( C=90 \% \) if \( \sigma=7.21 \) and \( E=2 \) then Sample Size

Upstudy AI Solution

Answer

Solution

Beyond the Answer

Related Questions

Latest Statistics Questions