Answer:
The sample size should be n=416 students.
Step-by-step explanation:
Hello!
The researchers wish to know which sample size should he take to estimate the true average number of alcoholic drinks all FSU undergraduate students who are members of a fraternity or sorority have in one week period.
The study variable is then
X: Number of drinks per week an undergraduate sorority/fraternity member has.
Preliminary studies show that the standard deviation is σ= 2.6 drinks per week.
Using a 1 - α: 0.95 of probability you have to find the sample size to estimate the population mean for a confidence interval with a margin error of no more than d= 0.25
Assuming that the variable has a normal distribution, the best statistic to use for the confidence interval is the standard normal, then the formula for the interval is:
X[bar] ± [tex]Z_{1-\alpha /2}[/tex] * [tex]\frac{Sigma}{\sqrt{n} }[/tex]
Where the margin of error is:
d= [tex]Z_{1-\alpha /2}[/tex] * [tex]\frac{Sigma}{\sqrt{n} }[/tex]
Using the given indormation you have to clear the sample size:
[tex]\frac{d}{ Z_{1-\alpha /2}}[/tex]= [tex]\frac{Sigma}{\sqrt{n} }[/tex]
[tex](\sqrt{n} )*(\frac{d}{ Z_{1-\alpha /2}})[/tex]= Sigma
[tex]\sqrt{n}[/tex]= [tex]\frac{Z_{1-\alpha /2}*Sigma}{d}[/tex]
n= [tex](\frac{Z_{1-\alpha /2}*Sigma}{d})^2[/tex]
n= [tex](\frac{1.965*2.6}{0.25})^2[/tex]= 415.50
Now since you cannot take a sample of 415.50 students, you have to round it to the next integer, so the sample size the researcher should take is 416 students.
I hope it helps!
