Question

19.

Are you smarter than your older brother? In a study of birth order and intelligence, IQ tests were given to 18- and 19-year-old men to estimate the size of the difference, if any, between the mean IQs of firstborn sons and of secondborn sons. The following data for 10 firstborn sons and 10 secondborn sons are consistent with the means and standard deviations reported in the article. It is reasonable to assume that the samples come from populations that are approximately normal. Firstborn 104 82 102 96 129 89 114 107 89 103 Secondborn 103 103 91 113 102 103 92 90 114 113 a. Construct a 95% confidence interval for the difference in mean IQ between firstborn and secondborn sons. b. Based on the confidence interval, is it reasonable to believe that firstborn sons and secondborn sons may have the same mean IQ?

Answer 1

	Values ( X )	Σ ( Xi- X̅ )2	Values ( Y )	Σ ( Yi- Y̅ )2
	104	6.25	103	0.36
	82	380.25	103	0.36
	102	0.25	91	129.96
	96	30.25	113	112.36
	129	756.25	102	0.16
	89	156.25	103	0.36
	114	156.25	92	108.16
	107	30.25	90	153.76
	89	156.25	114	134.56
	103	2.25	113	112.36
Total	1015	1674.5	1024	752.4

Mean X̅ = Σ Xi / n
X̅ = 1015 / 10 = 101.5
Sample Standard deviation S_X = √ ( (Xi - X̅ )² / n - 1 )
S_X = √ ( 1674.5 / 10 -1 ) = 13.6402

Mean Y̅ = ΣYi / n
Y̅ = 1024 / 10 = 102.4
Sample Standard deviation S_Y = √ ( (Yi - Y̅ )² / n - 1 )
S_Y = √ ( 752.4 / 10 -1) = 9.1433

Part a)

Confidence interval :-

t(α/2, DF) = t(0.05 /2, 15 ) = 2.131

(#1 – X2) ta/2,DF (S?/n1) + (S}/n2)

DF = ((S/n1+ s/n2))/((S/nl)/n1 - 1) + (S/n2)/n2 - 1))


DF = ((13.6402/10+9.1433/10))/((13.64022/10) /10-1)+(9.1433/10) /10-

DF = 15

(101.5 – 102.4) – t0.05/2,15 (13.64022/10) + (9.14332/10)

Lower Limit =
(101.5 – 102.4) – t005/2,15 (13.64022/10) + (9.14332/10)

Lower Limit = -11.9659
Upper Limit =
(101.5 – 102.4) + 0.05/2,15 (13.64022/10) + (9.14332/10)

Upper Limit = 10.1659
95% Confidence interval is ( -11.9659 , 10.1659 )

Part b)

Since the interval contain 0, we can conclude that  firstborn sons and secondborn sons may have the same mean IQ.