To determine Jose's margin of error, we need to follow these steps individually.
Step 1: Identify the Values Provided
- Standard deviation (Std Dev) for Jose's sample: [tex]\( \sigma = 50 \)[/tex] pounds
- Sample size for Jose's sample: [tex]\( n = 25 \)[/tex]
Step 2: Margin of Error Formula
The formula to calculate the margin of error (ME) is given:
[tex]\[ \text{ME} = 1.96 \times \left( \frac{\sigma}{\sqrt{n}} \right) \][/tex]
where:
- [tex]\( 1.96 \)[/tex] is the z-value for a 95% confidence interval,
- [tex]\( \sigma \)[/tex] is the standard deviation,
- [tex]\( n \)[/tex] is the sample size.
Step 3: Plug in the Values
Using the values provided for Jose's sample:
[tex]\[ \text{ME} = 1.96 \times \left( \frac{50}{\sqrt{25}} \right) \][/tex]
Step 4: Calculate the Standard Error
First, compute the denominator:
[tex]\[ \sqrt{25} = 5 \][/tex]
Now, substitute it back into the formula:
[tex]\[ \text{ME} = 1.96 \times \left( \frac{50}{5} \right) \][/tex]
Step 5: Simplify the Expression
[tex]\[ \frac{50}{5} = 10 \][/tex]
Thus:
[tex]\[ \text{ME} = 1.96 \times 10 = 19.6 \][/tex]
Step 6: Round to the Nearest Whole Number
Finally, round 19.6 to the nearest whole number:
[tex]\[ \text{Rounded ME} = 20 \][/tex]
Therefore, Jose's margin of error, rounded to the nearest whole number, is:
(C) 20
