Is there a difference in the gestation time for first and second births to the s

Question

Is there a difference in the gestation time for first and second births to the same parent? The following table shows the gestation time (days) for a sample of four mothers with their first two children.
Mother 1 2 3 4 First birth 245 267 249 263 Second birth 243 267 243 261

Assume that the changes in gestation (second time—first time) have an approximately Normal distribution. A 90% confidence interval for the mean difference in gestation times between the first and second births is –2.5 ± 4.00. –2.5 ± 2.96 –2.5 ± 5.92. –2.5 ± 2.07. Is there a difference in the gestation time for first and second births to the same parent? The following table shows the gestation time (days) for a sample of four mothers with their first two children.
Mother 1 2 3 4 First birth 245 267 249 263 Second birth 243 267 243 261

Assume that the changes in gestation (second time—first time) have an approximately Normal distribution. A 90% confidence interval for the mean difference in gestation times between the first and second births is –2.5 ± 4.00. –2.5 ± 2.96 –2.5 ± 5.92. –2.5 ± 2.07.

Explanation / Answer

following information has been generated using the given data

we use t-test for testing of difference in the gestation time for first and second births to the same parent and

t=(mean1-mean2)/((sp*(1/n1 +1/n2)1/2) =(256-253.5)/(11.54*sqrt(1/4 + 1/4))=0.3

and sp2=((n1-1)s12+(n2-1)s22)/n and with df is n=n1+n2-2

sp2=((n1-1)s12+(n2-1)s22)/n =(3*113.3+3*153)/(4+4-2)=133.15

sp=11.54

critical t(0.1,6)=1.94 is more than the calculated t=0.3, so there is no difference between gestation time for first and second births

difference =second - first=253.5-256=-2.5

SE(difference)=((sp*(1/n1 +1/n2)1/2)=(11.54*sqrt(1/4 + 1/4))=8.16

since sample size is small(<30) we use t-distribution for calculating the confidence interval

(1-alpha)*100% confidence interval for difference of mean=difference±t(alpha/2,n-1)*SE(difference)

=-2.5±1.94*8.16=-2.5±15.83

Mother 1 2 3 4 mean s2 n First birth 245 267 249 263 256 113.3333 4 Second birth 243 267 243 261 253.5 153 4

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