Question

two owners of a cattle ranch, jose and valeria, want to find the average weight for the ranchs 200 cows. instead of weighing all of the cows: jose weighs 25 cows and gets an average weight of 1,350 lb. (stdev 50) valeria weighs 100 cows and gets an average weight of 1,420 lb. (stdev 50) what is joses margin of error rounded to the nearest whole number? (the formula is) a 529 b 20 c 50 d 10

Explanation:

Step1: Recall the formula for the margin of error for a sample mean

The formula for the margin of error $E$ when the population standard - deviation $\sigma$ is known is $E = z_{\alpha/2}\frac{\sigma}{\sqrt{n}}$. In a large - sample situation (when $n$ is reasonably large, we can approximate the critical value $z_{\alpha/2}$ for a 95% confidence interval as $z_{\alpha/2}=1.96\approx2$. Here, we will use the combined sample information. First, we need to use the formula for the pooled standard - deviation (since we assume the variances are equal in a sense for the purpose of estimating the overall mean), but an alternative and simpler approach for a quick estimate of the margin of error when we want to estimate the population mean of all 200 cows is to consider the standard - deviation and sample size information. We'll use the fact that for a sample mean, the margin of error formula gives us an idea of the uncertainty. Since we don't have information about the confidence level, we'll make a rough estimate assuming a 95% confidence level. For a sample, the margin of error is related to the standard - deviation and the square root of the sample size.

Step2: Calculate the margin of error for Jose's sample

For Jose's sample, $n_1 = 25$ and $\sigma_1=50$. Using the formula $E_1=z_{\alpha/2}\frac{\sigma_1}{\sqrt{n_1}}$. Assuming $z_{\alpha/2} = 2$, we have $E_1=2\times\frac{50}{\sqrt{25}}$.
$E_1=2\times\frac{50}{5}=20$.

Answer:

B. 20