question archive Never forget that even small effects can be statistically significant if the samples are large
Never forget that even small effects can be statistically significant if the samples are large
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Never forget that even small effects can be statistically significant if the samples are large. To illustrate this fact, consider a sample of 147 small businesses. During a three-year period, 14 of the 100 headed by men and 8 of the 47 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 a = 0.05)? The P-value was so we conclude that The test showed no significant difference. (b) Now suppose that the same sample proportion came from a sample 30 times as large. That is, 240 out of 1410 businesses headed by women and 420 out of 3000 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 so we conclude that The test showed strong evidence of a significant difference. (0) 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
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a )
P-value was 0.6319 , so we conclude that , we cannot conclude test
b )
P-value was 0.0087 , so we conclude that , we can conclude the test . reject the null hypothesis .
c )
Interval for smaller samples : -0.1539 to 0.0935
Interval for larger samples : -0.0528 to -0.0076
Step-by-step explanation
a ) To find the P-value ,
Given that ,
Sample size , of men and women failed is ,
n1?=100 and n2?=47
and the variable x ,
x1?=14 and x2?=8
The sample proportions are ,
P1?=n1?x1??=10014?=0.14
P2?=n2?x2??=478?=0.1702
There fore ,
P?=n1?+n2?x1?+x2??=100+4714+8?=14722?=0.1497
The z value is,
z=P?(1−P?)(n1?1?+n2?1??)P1?−P2??
=0.1497×(1−0.1497)×(1001?+471?)?0.14−0.1702?
=0.003981195057?−0.0302?
z = -0.4786303291 ≈ - 0.479
So the z value is -0.4786
Given ,
Significance level , α=0.05 and it is two sided .
Image transcription text
P Value from Z Score Calculator This is very easy: just stick your Z score in the box marked Z score, select your significance level and whether you're testing a one or two-tailed hypothesis (if you're not sure, go with the defaults), then press the button! If you need to derive a Z score from raw data, you can find a Z test calculator here. Z score: -0.479 Significance Level: O0.01 O0.05 O0.10 One-tailed or two-tailed hypothesis?: Oone-tailed O Two-tailed The P-Value is .631939. The result is not significant at p <
.05.
Therefore ,
The P-value is ,
P-value = 0.631939 ≈ 0.6319
P-value is greater than 0.05 ( p-value > 0.05 ) . so , we cannot conclude test
P-value was 0.6319 , so we conclude that , we cannot conclude test
b ) To find the P-value ,
Given that ,
Sample size , of men and women fail respectively , is ,
n1?=3000 and n2?=1410
and the variable x of women and men respectively ,
x1?=420 and x2?=240
The sample proportions are ,
P1?=n1?x1??=3000420?=0.14
P2?=n2?x2??=1410240?=0.1702
There fore ,
P?=n1?+n2?x1?+x2??=3000+1410420+240?=4410660?=0.1497
The z value is,
z=P?(1−P?)(n1?1?+n2?1??)P−?P2??
=0.1497×(1−0.1497)×(14101?+30001?)?0.14−0.1702?
=0.0001327065019?−0.0302?
z = -2.62156628 ≈ -2.622
So the z value is 2.622
Given ,
Significance level , α=0.05 and it is two sided .
Image transcription text
P Value from Z Score Calculator This is very easy: just stick your Z score in the box marked Z score, select your significance level and whether you're testing a one or two-tailed hypothesis (if you're not sure, go with the defaults), then press the button! If you need to derive a Z score from raw data, you can find a Z test calculator here. Z score: -2.622 Significance Level: O0.01 O0.05 O0.10 One-tailed or two-tailed hypothesis?: OOne-tailed O Two-tailed The P-Value is .008742. The result is significant at p
<.05.
Therefore the P-value = 0.0087 .
P-value is less than 0.05 ( p-value < 0.05 ) . so Reject null hypothesis . we can conclude this test .
P-value was 0.0087 , so we conclude that , we can conclude the test . reject the null hypothesis .
c )
Interval for smaller samples ,
Sample size , of men and women failed is ,
n1?=100 and n2?=47
and the variable x ,
x1?=14 and x2?=8
The sample proportions are ,
P1?=n1?x1??=10014?=0.14
P2?=n2?x2??=478?=0.1702
There fore ,
P?=n1?+n2?x1?+x2??=100+4714+8?=14722?=0.1497
95% confidence interval is ,
(P1?−P2?)±zα/2?P(1−P)(n1?1?+n2?1?)?
Critical z value for 95% confidence interval is 1.96 .
(0.14−0.1702)±1.96×0.1497×(1−0.1497)×(1001?+471?)?
( -0.0302 - 0.1237 , -0.0302 + 0.1237 )
( -0.1539 , 0.0935 )
Therefore , Interval for smaller samples : -0.1539 to 0.0935
Interval for smaller larger ,
Given that ,
Sample size , of men and women fail respectively , is ,
n1?=3000 and n2?=1410
and the variable x of women and men respectively ,
x1?=420 and x2?=240
The sample proportions are ,
P1?=n1?x1??=3000420?=0.14
P2?=n2?x2??=1410240?=0.1702
There fore ,
P?=n1?+n2?x1?+x2??=3000+1410420+240?=4410660?=0.1497
95% confidence interval is ,
(P1?−P2?)±zα/2?P(1−P)(n1?1?+n2?1?)?
Critical z value for 95% confidence interval is 1.96 .
(0.14−0.1702)±1.96×0.1497×(1−0.1497)×(30001?+14101?)?
( -0.0302 - 0.0226 , -0.0302 + 0.0226 )
( -0.0528 , -0.0076 )
Therefore , Interval for larger samples : -0.0528 to -0.0076
