A simple random sample of 45 elements has been selected from a population of size 800. The sample mean is 600, and the sample standard deviation is 60. An approximate 95% confidence interval for the population total is
A) 583 to 617
B) 466,400 to 493,600
C) 400,000 to 500,000
D) 350,000 to 450,000

Question

Quizplus · Accepted Answer

To find the 95% confidence interval for the population total, we first find the confidence interval for the mean and then multiply the endpoints by the population size (N). The formula for a 95% confidence interval for the mean is $\bar{x} \pm z \frac{s}{\sqrt{n}}$, where $\bar{x}$ is the sample mean, $s$ is the sample standard deviation, $n$ is the sample size, and $z$ is the z-score corresponding to the desired level of confidence (approximately 1.96 for 95%).Given: $\bar{x} = 600$, $s = 60$, $n = 45$, and $N = 800$.First, calculate the standard error of the mean (SEM): $SEM = \frac{s}{\sqrt{n}} = \frac{60}{\sqrt{45}} \approx 8.944$.Then, calculate the margin of error (ME): $ME = z \times SEM = 1.96 \times 8.944 \approx 17.53$.Next, find the confidence interval for the mean: $600 \pm 17.53$, which gives us $582.47$ to $617.53$.Finally, to find the confidence interval for the total, multiply each endpoint by the population size (800): $582.47 \times 800 = 465,976$ and $617.53 \times 800 = 494,024$.Rounding to the nearest hundred gives us $466,400$ to $493,600$, which matches option B.

A Simple Random Sample of 45 Elements Has Been Selected