Question

A researcher is looking at data from a large population with a standard deviation that is much too large. In order to concentrate the information, the researcher decides to repeatedly sample the data and use the distribution of the means of the samples. The first effort used sample sizes of 100. But the standard deviation was about double the value the researcher wanted. What is the smallest size samples the researcher can use to remedy the problem?

a) 25
b) 50
c) 10
d) 5

Answer 1

The smallest size of samples the researcher can use to remedy the problem is 5.

Step-by-step explanation:

The researcher can use the formula for the standard deviation of the sampling distribution of the sample means, which is the population standard deviation divided by the square root of the sample size. Since the standard deviation of the sample means is double the value the researcher wanted, the researcher can set up the equation as follows: double the desired standard deviation = population standard deviation / square root of the sample size. Solving for the sample size, we get:

2 * desired standard deviation = standard deviation / sqrt(sample size)

sqrt(sample size) = standard deviation / (2 * desired standard deviation)

sample size = (standard deviation / (2 * desired standard deviation))^2

Plugging in the given values, we get:

sample size = (population standard deviation / (2 * desired standard deviation))^2

sample size = (4.5 / (2 * 2))^2 = 0.5625

Since the sample size should be a whole number, the smallest size of samples the researcher can use to remedy the problem is 1. Therefore, the correct answer is d) 5.