Question

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10. 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 20 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	44.602	53.256
2	35.697	54.993
3	37.516	62.523
4	36.726	51.003
5	32.282	53.036
6	32.834	61.879
7	41.561	43.774
8	47.027	55.129
9	40.814	59.171
10	42.289	45.363
11	37.424	49.436
12	38.557	57.767
13	41.178	52.624
14	38.226	47.491
15	43.388	47.562
16	33.950	52.447
17	40.587	45.644
18	43.206	53.841
19	42.385	42.281
20	47.313	45.463

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. 7.86 to 15.86 ng/mL

b. 6.99 to 18.39 ng/mL

c. 8.76 to 17.84 ng/mL

d. 6.68 to 14.27 ng/mL

Answer 1

for 95% CI; and 19 degree of freedom, value of t=			2.093
therefore confidence interval=sample mean -/+ t*std error
margin of errror          =t*std error=		4.000451
lower confidence limit                     =		7.86
upper confidence limit                    =		15.86
from above 95% confidence interval for population mean =(7.86 ,15.86)

option A is correct