Question

PLEASE SHOW ALL WORKING STEPS. THANKS

8. 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 30 landscapers and 50 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	52.732
2	55.072
3	49.285
4	53.121
5	50.824
6	50.681
7	58.735
8	58.919
9	55.206
10	50.427
11	56.894
12	53.921
13	56.994
14	52.553
15	52.844
16	59.129
17	51.895
18	52.775
19	43.229
20	44.138
21	48.417
22	48.871
23	57.559
24	56.529
25	54.408
26	64.301
27	51.168
28	46.713
29	47.306
30	49.357

Table 2. Office Workers
Subject	Vitamin D
1	33.588
2	29.509
3	41.073
4	31.099
5	34.916
6	35.989
7	38.288
8	33.433
9	36.057
10	32.349
11	31.597
12	27.334
13	26.669
14	27.254
15	28.483
16	28.601
17	27.722
18	28.376
19	28.135
20	37.940
21	28.275
22	41.984
23	38.529
24	36.131
25	30.104
26	36.741
27	36.456
28	32.489
29	25.914
30	28.119
31	29.942
32	31.745
33	36.530
34	32.565
35	28.298
36	32.281
37	35.371
38	26.921
39	36.623
40	31.120
41	27.441
42	23.342
43	28.873
44	29.357
45	27.941
46	32.926
47	32.529
48	35.324
49	30.805
50	35.566

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. 17.32 to 25.79 ng/mL

b. 20.99 to 25.68 ng/mL

c. 18.83 to 22.82 ng/mL

d. 20.15 to 26.25 ng/mL

Answer 1

c. 18.83 to 22.82 ng/mL

The formula is:

The output is:

Landscapers	Office Workers
52.80010	31.97308	mean
4.70179	4.23284	std. dev.
30	50	n
	78	df
	20.83	difference (Landscapers - Office Workers)
	19.47	pooled variance
	4.41	pooled std. dev.
	1.02	standard error of difference
	18.83	confidence interval 95.% lower
	22.82	confidence interval 95.% upper
	2.03	margin of error