Question

Recall in our discussion of the normal distribution the research study that examined the blood vitamin D levels of the entire US population of landscape gardeners. The intent of this large-scale and comprehensive study was to characterize fully this population of landscapers as normally distributed with a corresponding population mean and standard deviation, which were determined from the data collection of the entire population.

Suppose you are now in a different reality in which this study never took place though you are still interested in studying the average vitamin D levels of US landscapers. In other words, the underlying population mean and standard deviation are now unknown to you. Furthermore, you would like to collect data from US office workers to examine the difference between the average vitamin D levels of landscapers and office workers, which will reflect any occupational sun exposure differences as measured by blood vitamin D levels. You obtain research funding to sample at random 32 landscapers and 32 office workers, collect blood samples, and send these samples to your collaborating lab in order to quantify the amount of vitamin D in both groups' blood. After anxiously awaiting your colleagues to complete their lab quantification protocol, they email you the following vitamin D level data as shown in the following tables.

Table 1. Landscapers

Subject	Vitamin D
1	43.104
2	45.147
3	43.588
4	46.764
5	43.083
6	50.448
7	39.430
8	48.074
9	40.328
10	44.983
11	39.339
12	39.977
13	43.447
14	40.712
15	51.308
16	42.451
17	42.774
18	54.254
19	44.717
20	40.029
21	31.471
22	38.238
23	47.022
24	41.168
25	35.647
26	44.090
27	45.703
28	52.699
29	46.258
30	46.365
31	53.494
32	44.693

Table 2. Office Workers

Subject	Vitamin D
1	28.820
2	29.225
3	36.528
4	33.125
5	33.972
6	41.400
7	32.249
8	32.337
9	32.130
10	28.187
11	40.917
12	34.138
13	38.358
14	30.696
15	32.616
16	30.722
17	33.495
18	32.288
19	33.601
20	30.804
21	35.609
22	34.327
23	28.659
24	36.964
25	40.766
26	37.309
27	34.924
28	36.628
29	35.763
30	29.431
31	34.400
32	23.855

What is the estimated 95% confidence interval (CI) of the average difference in blood vitamin D levels between US landscapers and office workers in ng/mL? Assign groups 1 and 2 to be landscapers and office workers, respectively.

Please note the following: 1) in practice, you as the analyst decide how to assign groups 1 and 2 and subsequently interpret the results appropriately in the context of the data, though for the purposes of this exercise the groups are assigned for you; 2) you might calculate a CI that is different from any of the multiple choice options listed below due to rounding differences, therefore select the closest match; 3) ensure you use either the large or small sample CI formula as appropriate; and 4) you may copy and paste the data into Excel to facilitate analysis.

Select one:

a. 7.47 to 14.01 ng/mL

b. 7.43 to 14.71 ng/mL

c. 8.30 to 12.74 ng/mL

d. 6.97 to 10.95 ng/mL

Answer 1

	table 1		table 2
x1    =	44.09	x2      =	33.57
s1    =	5.04	s2      =	3.95
n1    =	32	n2     =	32
Pooled Variance Sp2=((n1-1)s21+(n2-1)*s22)/(n1+n2-2)=				20.5338

standard error se =Sp*√(1/n1+1/n2)=

1.133

point estimate of difference=x1-x2=			10.518
for 95 % CI & 62 df value of t=			1.999
margin of error E=t*std error =			2.265
lower bound=mean difference-E=			8.253
Upper bound=mean differnce +E=			12.782
from above 95% confidence interval for population mean =(8.253,12.782)

Correct option is C:  8.30 to 12.74 ng/mL