Question

(10 pts) Archeologists have discovered a sample of ancient Egyptian skulls from two different time periods, 4000 B.C. and 150 A.D. They believe that the mean maximum skull breadth (the width of the skull at its widest part) has increased from 4000 B.C. to 150 A.C. A sample of 13 Eqyptian skull breadths (in mm) from each time period is show below:

4000 B.C.

124, 138, 123, 118, 136, 134, 138. 135, 121. 135, 120. 136, 118

150 A.D.

(a) Calculate the mean and standard deviation for each data set. Round each mean to 2 decimal places, and each standard deviation to 4 decimal places.

4000 B.C. mean: 150 A.D. mean:

4000 B.C. standard deviation: 150 A.D. standard deviation:

(b) Using your answers from part (a) and the formula for a 95% confidence interval as presented in lecture, fill in the blanks with the appropriate values for this problem for calculating the confidence interval below. To enter $\sqrt x$ where x is any number, type sqrt(x). For example, should be written as sqrt(2).

4000 B.C. skull breadths:

-

to        +

150 A.D. skull breadths:

-

to        +

(c) Using the formulas from part (b), find the 95% confidence intervals for the mean maximum skull breadths of all Egyptian skulls from each period. Round each answer to the nearest hundredth (2 decimal places).

4000 B.C. skull breadths:

to

150 B.C. skull breadths:

to

(d) Based on these confidence intervals, can we conclude with 95% certainty that the true population mean skull breadth of the Egyptian male has changed from 4000 B.C. to 150 A.D.?

A. Yes. The confidence intervals for skull breadths from 4000 B.C. and 150 A.D. do not overlap, so the true population mean skull breadth for the two time periods could be different.
B. Yes. The true population mean skull breadth for 150 A.D. skulls must be higher than that 4000 B.C. skulls since the confidence interval limits are higher for the 150 A.D. skulls.
C. No. The margin of error of the confidence intervals for 150 a.D. skulls and 4000 B.C. skulls are unequal, so the true population mean skull breadths for the two time periods could be the same.
D. No. The confidence intervals for skull breadths from 4000 B.C. and 150 A.D. overlap, so it's possible that the true population mean skull breadth for the two time periods could be the same.

Answer 1

Calculations for mean and standard deviation:

4000B.c2 15376 19044 15129 13924 18496 17956 19044 18225 14641 18225 14400 18496 13924 216880 4000B.C 124 138 123 136 134 138 135 121 135 120 136 Sum 1676 The sample mean X is computed as follows 16761 128.923 13 Also, the sample variance s is 13-1 21688 1676 Therefore, the sample sandartd deviation s is s = V82-V67:077 = 8.19

150A.D 139 131 138 138 133 137 134 141 142 127 133 134 130 1757 150A.D2 19321 17161 19044 19044 17689 18769 17956 19881 20164 16129 17689 17956 16900 237703 Sum The sample mean X is computed as follows 1757 135.154 72 13 Also, the sample variance s is 175 13 ) = 19.808 ( 237703- = 13-1 Therefore, the sample sandartd deviation s is v19.808 4.451
4000 B.C. mean:128.923

150 A.D. mean:135.154

4000 B.C. standard deviation:8.19

150 A.D. standard deviation:4.451

Confidence Interval Calculations:

4000BC

The provided sample mean is X = 128.923 and the sample standard deviation is s = 8.19. The size of the sample is n-13 and the required confidence level is 95%. The number of degrees of freedom are df = 13-1 = 12, and the significance level is α = 0.05, Based on the provided information, the critical t-value for a = 0.05 and df = 12 degrees of freedom is te 2.179, The 95% confidence for the population mean μ is computed using the following expression Therefore, based on the information provided, the 95 % confidence for the population mean is cr = (128.923 2.179 x 8.19, 1282219 8.19 CI 128.923 128.923- ν'13 - (128.923-4.949, 128.923+4.949) (123.974, 133.872) which completes the calculation.

150 A.D

Solution: The provided sample mean is = 135.154 and the sample standard deviation is s = 4.451. The size of the sample is n-13 and the required confidence level is 95%. The number of degrees of freedom are df = 13-1-12, and the significance level is α = 0.05. Based on the provided information, the critical t-value for α = 0.05 and df 12 degrees of freedom is te 2.179. The 95% confidence for the population mean μ is computed using the following expression Vm Therefore, based on the information provided, the 95 % confidence for the population mean μ is 135,154,135.15421 6 2.179 x 4.451 2.179 x 4.451 CI 135.154 = (135.154-2.69, 135.154 + 2.69) (132.464.137.844) which completes the calculation.

d) Based on these confidence intervals, can we conclude with 95% certainty that the true population mean skull breadth of the Egyptian male has changed from 4000 B.C. to 150 A.D.?

D. No. The confidence intervals for skull breadths from 4000 B.C. and 150 A.D. overlap, so it's possible that the true population mean skull breadth for the two time periods could be the same.