Sampling Distribution
Complete solutions and illustrations are required.
5.The standard deviation of a variable is 15. If a sample of 100 individuals are selected compute for
a.The standard error of the mean.
b.What sample size is necessary to double the standard error of the mean?
c.What sample size is needed to cut the standard error of the mean in half?
You seem to be homework dumping.
To answer these questions, we first need to understand the concept of sampling distribution and its relationship with standard deviation and standard error of the mean.
A sampling distribution is a distribution that shows the frequencies or probabilities of different outcomes that could occur when samples are taken from a given population. It provides information about the variability of sample means.
The standard deviation measures the variability or spread of a dataset. It quantifies how much the values in a dataset differ from the mean.
The standard error of the mean (SEM) measures the precision or accuracy of the sample mean as an estimate of the population mean. It represents the standard deviation of the sampling distribution of sample means.
Now, let's answer the given questions:
a. To compute the standard error of the mean, we can use the formula:
SEM = standard deviation / sqrt(sample size)
In this case, the standard deviation is given as 15. And the sample size is 100.
SEM = 15 / sqrt(100)
SEM = 15 / 10
SEM = 1.5
Therefore, the standard error of the mean is 1.5.
b. To find the sample size necessary to double the standard error of the mean, we can rearrange the formula for SEM:
sample size = (standard deviation / SEM)^2
We want to double the SEM, so we need to find the sample size when SEM is doubled. Let's say the new SEM is 2 times the original SEM.
sample size = (15 / (1.5 * 2))^2
sample size = (15 / 3)^2
sample size = 5^2
sample size = 25
Therefore, a sample size of 25 is necessary to double the standard error of the mean.
c. To find the sample size needed to cut the standard error of the mean in half, we can again rearrange the formula for SEM:
sample size = (standard deviation / SEM)^2
We want to reduce the SEM by half, so we need to find the sample size when SEM is reduced by half. Let's say the new SEM is half of the original SEM.
sample size = (15 / (1.5 / 2))^2
sample size = (15 / 0.75)^2
sample size = 20^2
sample size = 400
Therefore, a sample size of 400 is needed to cut the standard error of the mean in half.