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	1.329	1.338	1.075	1.229	0.937	1.006	1.179	1.287
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=________

Describe the distribution of these differences using words.

___The distribution is uniform.

___The distribution is left skewed.    

___The sample is too small to make judgments about skewness or symmetry.

___The distribution is Normal.The distribution is right skewed.

(b) 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.) ___________

Give your conclusion. (Use the significance level of 5%.)

____We cannot reject H₀ based on this sample.

____We can reject H₀ based on this sample.    

(c) 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.)

(______________ , _____________ )

  ,

(d) The eight subjects used for this comparison were not a random sample. In fact, they were friends of the researchers whose ages and weights were similar to the types of people who would be measured with this DXA machine. Comment on the appropriateness of this procedure for selecting a sample, and discuss any consequences regarding the interpretation of the significance-testing and confidence interval results.

____The subjects from this sample may be representative of future subjects, but the test results and confidence interval are suspect because this is not a random sample.

_____The subjects from this sample, test results, and confidence interval are representative of future subjects.

  

Answer 1

(a1) =mean=-0.0015 and s=sample standard deviation=0.0106

(a2)The sample is too small to make judgments about skewness or symmetry

(b1) t=-0.4004,

(b2)degree of freedom=n-1=8-1=7

(b3)p-value=0.7008 , this is two tailed test as the null hypothesis H0:=0 and alternate hypothesis Ha:  

(b4) We cannot reject H₀ based on this sample,

as the p-value =0.7008 is more than level of significance=alpha=0.05

(c)  required (1- α)*100% confidence interval =||±t(α/2,n-1)*SE()

95% confidence interval=0.0015±2.3646*0.0106=0.0015±0.0251=(-0.0236,0.0266)

(d) The subjects from this sample may be representative of future subjects, but the test results and confidence interval are suspect because this is not a random sample.

One of the assumption of statistical hypothesis test is the sample should be random, here it is violation of this assumption

following information has been generated

s.n.	operrator1	operrator2	d			n=	8
1	1.329	1.323	0.006			mean=d_bar	-0.0015
2	1.338	1.322	0.016			sd=	0.0106
3	1.075	1.073	0.002			SE(d_bar)=sd/sqrt(n)=	0.0037
4	1.229	1.233	-0.004			t=d_bar/SE(d_bar)=	-0.4004
5	0.937	0.934	0.003		one tailed	critical t=	1.8946
6	1.006	1.019	-0.013		two tailed	critical t=	2.3646
7	1.179	1.184	-0.005		two tailed	p-value=	0.7008
8	1.287	1.304	-0.017		one tailed	p-value=	0.3504
		mean=	-0.0015
		s=	0.0106