Could Someone Please Help With This Statistics Problem?

Question

A group of researchers is studying the relationship between cortisol (stress hormone) levels and memory, and they want to see if a sample of 100 adults that has been recruited is a good representation of the population it came from, before they conduct additional research. The population has been found to be normally distributed and have a mean cortisol level of 12 mcg/dL, with a standard deviation of 2 mcg/dL. The sample was found to have a mean cortisol level of 15 mcg/DL, with a standard deviation of 3 mcg/dL.

For this scenario, construct a confidence interval to determine if this sample mean is significantly different from the population mean. Explain how you know, based on the confidence interval, and specify the confidence level you used. 

Answer 1

Hi Sarah,

While we know the true population mean and standard deviation, to get the comparison, we have to use the sample mean and standard deviation to get the confidence interval. We know data are normally distributed and we know a population standard deviation, even though we don't use it, so we use a z-confidence interval:

CI= xbar +/- z* SE

xbar=sample mean

z*=z-critical value

SE=s/sqrt(n)

s=sample standard deviation

n=sample size

Now, a 95% confidence interval is sufficient in most cases, and I don't know any difference in the medical world, so we'll use that. You can get the critical value from a z-table, but it is probably better to just memorize the most common:

90%: 1.645

95% 1.96

99%: 2.576

So, for our problem:

xbar = 15

z* = 1.96

s=3

n=100

SE= 3/sqrt(100)

SE=3/10

SE=0.3

CI= 15 +/- (1.96*0.3)

CI = (14.412, 15.588)

Is 12 in this interval? No. That means there is significant difference between sample mean and population mean. Be sure to specify that you used 95% confidence. I hope this helps.