Hi Salma,
First, we are looking at proportions, not means, so eliminate any answer choices that contain "mu." Second, the question said "smaller," so we are looking at the less than alternative, so only choice that makes sense is:
H0:pM=pFH0:pM=pF
H1:pM<pFH1:pM<pF
Less than also implies left-sided alternative.
Now, we have at least five successes (person owns cat) and failures (person does not) for both genders, so we can use z. Formula is:
z=(p-hat1-p-hat2)/sqrt(p0(1-p0)(1/n1 + 1/n2)
Breaking this down:
phat1=proportion of men who own cats in sample
phat2=proportion of women who own cats in sample
p0=combined proportion of cat owners for both gender samples
n1=men's sample size
n2=women's sample size
For this specific problem:
z=(p-hat1-p-hat2)/sqrt(p0(1-p0)(1/n1 + 1/n2)
phat1=0.25
phat2=0.5
n1=20
n2=60
p0 has its own formula, which is:
p0=(x1 + x2)/(n1 + n2)
x1=number of men who own cats
x2=number of women who own cats
n1=men's sample size
n2=women's sample size
x1=0.25*20= 5
x2=0.5*60= 30
n1=20
n2=60
p0=(5+30)/(20+60)
p0=35/80
p0=0.4375
Thus:
z=(p-hat1-p-hat2)/sqrt(p0(1-p0)(1/n1 + 1/n2)
phat1=0.25
phat2=0.5
p0=0.4375
n1=20
n2=60
z=(0.25 - 0.5)/sqrt[(0.4375*0.5625)((1/20)+ (1/60))]
z= -0.25/0.128
z= -1.95
That is your test statistic.
Critical value is z*= -2.576 or, to 2 decimals, z*= -2.58
That value may be worth memorizing; I didn't have a lot of 0.005, 1-sided alphas in intro. statistics, but it appears your instructor/text uses those provisions frequently. Anyhow, -1.95 is not less than -2.58. So we fail to reject the null and conclude no significant difference in proportion of men and women who own cats at the 0.005 significance level.
I hope this helps.
