Question

Assume, at MODUL university a representative sample of n=250 was drawn, 200 students were female.

What does this tell us about the percentage of female students studying at this university (the population)? (Hint: Calculate a 95%-confidence interval for your percentage prediction - namely the percentage value in the population.) (1p.)

Repeat the computation of the 95%-confidence interval with the following sample sizes: n=100, n=1500, and n=12000, each time assuming the same percentages as in 3). What happens and why? (1p.)

Answer 1

$z=\frac{p-p_0}{\sqrt{\frac{p_0(1-p_0)}{n}}}$

for n= 250 percentage of female students:

$p_0=200/250=4/5=0.8$

critical values for 95%-confidence interval:

$z=\pm 1.96$

$-1.96<\frac{p-0.8}{0.05}<1.96$

$0.70<p_0<0.90$

for n= 100:

$-1.96<\frac{p-0.8}{0.04}<1.96$

$0.72<p_0<0.88$

for n= 1500:

$-1.96<\frac{p-0.8}{0.01}<1.96$

$0.78<p_0<0.82$

for n= 12000:

$-1.96<\frac{p-0.8}{0.004}<1.96$

$0.793<p_0<0.807$

Increasing the sample size decreases the width of confidence intervals, because it decreases the standard error.