Question

Never forget that even small effects can be statistically significant if the samples are large. To illustrate this fact, consider a sample of 128 small businesses. During a three-year period, 12 of the 87 headed by men and 7 of the 41 headed by women failed.

(a) Find the proportions of failures for businesses headed by women and businesses headed by men. These sample proportions are quite close to each other. Give the P-value for the test of the hypothesis that the same proportion of women's and men's businesses fail. (Use the two-sided alternative). What can we conclude (Use α=0.05α=0.05)?
The P-value was  so we conclude that
Choose a conclusion. The test showed strong evidence of a significant difference. The test showed no significant difference.

(b) Now suppose that the same sample proportion came from a sample 30 times as large. That is, 210 out of 1230 businesses headed by women and 360 out of 2610 businesses headed by men fail. Verify that the proportions of failures are exactly the same as in part (a). Repeat the test for the new data. What can we conclude?
The P-value was _______ conclude that
Choose a conclusion:

The test showed strong evidence of a significant difference.

The test showed no significant difference.

(c) It is wise to use a confidence interval to estimate the size of an effect rather than just giving a P-value. Give 95% confidence intervals for the difference between proportions of men's and women's businesses (men minus women) that fail for the settings of both (a) and (b). (Be sure to check that the conditions are met. If the conditions aren't met for one of the intervals, use the same type of interval for both)
Interval for smaller samples: ___ to ____
Interval for larger samples: _________to _________

What is the effect of larger samples on the confidence interval?
Choose an effect:

. The confidence interval is unchanged.

The confidence interval's margin of error is reduced.

The confidence interval's margin of error is increased.

Answer 1

a)

x=	12	7
p̂=x/n=	0.1224	0.1707
n =	98	41
estimated prop. diff =p̂1-p̂2    =		-0.0483
pooled prop p̂ =(x1+x2)/(n1+n2)=		0.13669
std error Se=√(p̂1*(1-p̂1)*(1/n1+1/n2) =		0.0639
test stat z=(p̂1-p̂2)/Se =		-0.76
P value   =	0.4498	(from excel:2*normsdist(-0.76)

The test showed no significant difference.

b)

x=	360	210
p̂=x/n=	0.1379	0.1707
n =	2610	1230
estimated prop. diff =p̂1-p̂2    =		-0.0328
pooled prop p̂ =(x1+x2)/(n1+n2)=		0.14844
std error Se=√(p̂1*(1-p̂1)*(1/n1+1/n2) =		0.0123
test stat z=(p̂1-p̂2)/Se =		-2.67
P value   =	0.0076	(from excel:2*normsdist(-2.67)

since p value <0.05

The test showed strong evidence of a significant difference.

c)

since sample success is less than 10 for one sample. adding 1 in success and 2 in sample size

for smaller sample:

x=	13	8
p̂=x/n=	0.1461	0.1860
n =	89	43
estimated diff. in proportion=p̂1-p̂2=		-0.0400
Se =√(p̂1*(1-p̂1)/n1+p̂2*(1-p̂2)/n2) =		0.0702
for 95 % CI value of z=		1.960
margin of error E=z*std error =		0.137522
lower bound=(p̂1-p̂2)-E=		-0.1775
Upper bound=(p̂1-p̂2)+E=		0.0975
from above 95% confidence interval for difference in population proportion =(-0.1775 ,0.0975)

Interval for larger samples:

x=	361	211
p̂=x/n=	0.1382	0.1713
n =	2612	1232
estimated diff. in proportion=p̂1-p̂2=		-0.0331
Se =√(p̂1*(1-p̂1)/n1+p̂2*(1-p̂2)/n2) =		0.0127
for 95 % CI value of z=		1.960
margin of error E=z*std error =		0.024854
lower bound=(p̂1-p̂2)-E=		-0.0579
Upper bound=(p̂1-p̂2)+E=		-0.0082
95% confidence interval for difference in population proportion =(-0.0579 , -0.0082)

The confidence interval's margin of error is reduced.