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 examine if wearing tank tops instead of short sleeve shirts significantly effects vitamin D levels. To accomplish this, you propose to collect data from the landscapers at two different points in time. Specifically, the landscapers are to wear short sleeve shirts while outside working during a period of three weeks. After three weeks, you collect blood specimens and the landscapers are then to wear tank tops for the next three weeks under the same working conditions, after which you collect blood draws a second time. You obtain research funding to randomly sample 47 landscapers, collect blood samples at two different time points as described above, and send these samples to your collaborating lab in order to quantify the amount of vitamin D in the landscapers' 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 table.

Subject	Time Point 1, Shirts Vitamin D (ng/mL)	Time Point 2, Tank Tops Vitamin D (ng/mL)
1	34.330	30.788
2	19.610	37.881
3	31.042	23.725
4	33.297	32.672
5	31.087	36.612
6	26.439	37.082
7	36.062	35.038
8	38.644	27.368
9	34.605	33.235
10	31.443	39.501
11	32.047	30.624
12	32.922	37.986
13	34.154	34.025
14	47.880	35.900
15	25.695	25.717
16	27.708	37.217
17	31.614	39.901
18	47.791	35.699
19	31.065	31.782
20	26.677	30.591
21	33.730	24.477
22	17.734	34.588
23	37.498	34.670
24	31.232	34.012
25	38.163	35.626
26	30.603	28.071
27	43.433	28.679
28	31.998	30.565
29	34.906	28.796
30	25.123	33.729
31	28.010	33.014
32	38.383	34.753
33	35.078	39.856
34	43.383	27.229
35	29.450	37.744
36	36.372	36.058
37	29.220	32.527
38	38.142	33.627
39	34.887	33.287
40	18.731	34.337
41	33.886	29.764
42	17.781	27.027
43	37.166	34.641
44	33.491	37.149
45	24.732	35.403
46	33.113	33.931
47	46.122	36.412

What is the estimated 95% confidence interval (CI) of the average difference in blood vitamin D levels between short sleeve shirt and tank top attire amongst US landscapers in ng/mL?

Please note the following: 1) in practice, you as the analyst decide how to calculate the difference in vitamin D levels between time points for a given study participant, and subsequently interpret the aggregated results appropriately in the context of the data, though for the purposes of this exercise the difference is assigned for you as follows. Define the difference as the second minus the first time points, which is common practice, since the plus or minus sign of the resulting difference reflects any change over sequential time; 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. -1.60 to 3.14 ng/mL

b. -1.72 to 2.86 ng/mL

c. -1.99 to 3.32 ng/mL

d. -1.85 to 3.07 ng/mL

Answer 1

sample mean difference 'x̄=	0.571
sample size   n=	47.00
sample std deviation s=	8.00
std error 'sx=s/√n=	1.1671

for 95 % CI value of z=		1.96
margin of error E=z*std error =		2.29
lower bound=sample mean-E=		-1.72
Upper bound=sample mean+E=		2.86

correct option is b:

-1.72 to 2.86 ng/mL