Never forget that even small effects can be statistically significant if the sam

Question

Never forget that even small effects can be statistically significant if the samples are large. To illustrate this fact, consider a sample of 92 small business. During a three-year period, 8 of the 57 headed by men and 6 of the by women failed. 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 fall. Use the two-sided alternative). What can we conclude (Use alpha = 0.05)? The P-value was ____________ so we conclude that The test showed no significant difference. New suppose that the same sample proportion came from a sample 30 times as large. That is, 180 out of 1050 businesses headed by women and 240 out of 1710 businesses headed by men fall. verify that the proportions of failures are exactly as in part(a). Repeat the test for the new data What can we conclude? The P-value was ___________ So we conclude that The test showed strong evidence of a significant difference. 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 fall 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 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? The confidence interval margin of error is reduced.

Explanation / Answer

for small sample p1=8/57 =0.14 ; n1=57

p2=6/35=0.17 ' n2=35

hence std error of mean difference =(p1(1-p1)/n1+p2(1-p2)/n2)1/2 =0.0786

hence test stat =(p1-p2)/std error =-0.3955

for which p value 0.6925

for large sample p1=0.14 ; n1=1710

p2=0.17 ; n2=1050

hence std error =0.0143

test stat z =(p1-p2)/std error =-2.1662

for above p value =0.0303

as for 95% z=1.96

hence confidence interval for smaller sample =mean differnce +/- z*std error =-0.1851 ; 0.1229

for larger sample confidence ionterval =-0.0592 ; -0.0030

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