Question

Dual-energy X-ray absorptiometry (DXA) is a technique for measuring bone health. One of the most common measures is total body bone mineral content (TBBMC). A highly skilled operator is required to take the measurements. Recently, a new DXA machine was purchased by a research lab, and two operators were trained to take the measurements. TBBMC for eight subjects was measured by both operators. The units are grams (g). A comparison of the means for the two operators provides a check on the training they received and allows us to determine if one of the operators is producing measurements that are consistently higher than the other. Here are the data.

	Subject
Operator	1	2	3	4	5	6	7	8
1	1.327	1.335	1.076	1.226	0.939	1.005	1.180	1.286
2	1.323	1.322	1.073	1.233	0.934	1.019	1.184	1.304

(a)

Take the difference between the TBBMC recorded for Operator 1 and the TBBMC for Operator 2. (Use Operator 1 minus Operator 2. Round your answers to four decimal places.)

x=

s=

Use a significance test to examine the null hypothesis that the two operators have the same mean. Give the test statistic. (Round your answer to three decimal places.)

t =  

Give the degrees of freedom.

Give the P-value. (Round your answer to four decimal places.)

The sample here is rather small, so we may not have much power to detect differences of interest. Use a 95% confidence interval to provide a range of differences that are compatible with these data. (Round your answers to four decimal places.)

( , )

Answer 1

For operartor 1

= 1.172, s1 = 0.15

For operator 2

= 1.174, s2 = 0.149

H₀:

H₁:

THe test statistic t = ()/sqrt(s1^2/n1 + s2^2/n2)

                              = (1.172 - 1.174)/sqrt((0.15)^2/8 + (0.149)^2/8)

                              = -0.03

DF = (s1^2/n1 + s2^2/n2)^2/((s1^2/n1)^2/(n1 - 1) + (s2^2/n2)^2/(n2 - 1)

     = ((0.15)^2/8 + (0.149)^2/8)^2/(((0.15)^2/8)^2/7 + ((0.149)^2/8)^2/7)

     = 14

P-value = 2 * P(T < -0.03)

              = 2 * 0.4882

              = 0.9764

As the P-value is greater than the significance level (0.9764 > 0.05), so the null hypothesis is not rejected

So we can conclude that the two operators have the same mean.

At 95% confidence interval the critical value is t_{0.025, 14} = 2.145

The 95% confidence interval is

() +/- t_{0.025, 14} * sqrt(s1^2/n1 + s2^2/n2)

= (1.172 - 1.174) +/- 2.145 * sqrt((0.15)^2/8 + (0.149)^2/8)

= -0.002 +/- 0.1603

= -0.1623, 0.1583