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Two owners of a cattle ranch, Jose and Valeria, want to find the average weight of the ranch's 200 cows. Instead of weighing all of the cows:

Jose weighs 25 cows and gets an average weight of [tex]$1,350 \text{ lb}$[/tex] (stdev 50). Valeria weighs 100 cows and gets an average weight of [tex]$1,420 \text{ lb}$[/tex] (stdev 50).

What is Jose's margin of error rounded to the nearest whole number? (The formula is [tex]1.96 \times \frac{\text{StdDev}}{\sqrt{N}}[/tex])

A. 50
B. 529
C. 20
D. 10



Answer :

GinnyAnswer
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

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