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Some archaeologists theorize that ancient Egyptians interbred with several different immigrant populations over thousands of years. To see if there is any indication of the change in body structure that might have resulted, they measured a random sample of 30 skulls of male Egyptians dated from 4000 B.C.E and a random sample of 30 others dated from 200 B.C.E. and recorded the maximum skull breadth, in millimeters.

(a) (1 pt) What is the explanatory variable? , (b) (1 pt) What is the response variable?, (c) (1 pt) Define the parameters in the context of the problem.

(d) (4 pts) Check the conditions. Is it reasonable to conclude the responses of the skulls from 4000 B.C.E. are independent? Is it reasonable to conclude the responses of the skulls from 200 B.C.E. are independent? Is it reasonable to conclude the responses from skulls in 4000 B.C.E are independent of responses of skulls from 200 B.C.E.? Is it reasonable to conclude the sampling distribution of the difference in the average skull breadth is approximately normal?

(e) (2 pts) Obtain a 98% confidence interval for the difference in the mean skull breadth between these two eras.

(f) (4 pts) Conduct a formal test of the hypothesis to determine if the mean breadth of males’ skulls changed over this period. What are the null and alternative hypotheses? Determine the test statistic and p-value. What level of significance would correspond to the confidence interval in (e)? What is the conclusion? State the conclusion in the context of the problem.

Tabulate Skull Breadth 30 135.63333333 30 131.36666667 Era 200 BCE N Mean 4000 ВСЕ N Mean Std Dev 5.0572988034 Min 119 Мax 144 25 Range Variance 25.576271186 Std Err 0.6528944681 3.788238804 CV Median 133.5 Interquartile Range 7 Quantiles25 131 Quantiles50 133.5 Quantiles75 138

Answer 1

a) EXPLANATORY VARIABLE : Time : 4000 BCE and 200 BCE

b) RESPONSE VARIABLE: the change in body structure

c) the parameters in the context of the problem is mean of population from 4000BCE and mean of population from 200BCE

d)  It is reasonable to conclude the responses from skulls in 4000 B.C.E are independent of responses of skulls from 200 B.C.E.and  it is it reasonable to conclude the sampling distribution of the difference in the average skull breadth is approximately normal.

e) 98% confidence interval for the difference in the mean skull breadth between these two eras:

t critical=2.39

μ₁ - μ₂ = (M₁ - M₂) ± ts_{(M1 - M2)} = 4.26 ± (2.39 * 0.65) = 4.26 ± 1.55= (2.71 , 5.81)

f) NULL HYPOTHESIS H0:
μι = μ2

ALTERNATIVE HYPOTHESIS Ha:
ul + u2

t = x1bar-x2bar/S.E

= 4.26/0.65

t = 6.55

The p-value is 0.000.The result is significant because p < .02.

Decision: REJECT H0.

Level of significance would correspond to the confidence interval is 0.02

Conclusion: We have sufficient evidence to conclude that there is indication of the change in body structure.