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 10 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	27.841	41.044
2	23.758	40.676
3	22.552	44.099
4	35.153	43.491
5	27.075	36.704
6	32.630	37.067
7	26.739	43.767
8	31.131	27.602
9	33.666	43.961
10	19.705	39.774

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. 6.32 to 17.27 ng/mL

b. 5.31 to 15.37 ng/mL

c. 5.56 to 19.34 ng/mL

d. 5.53 to 15.28 ng/mL

Answer 1

		difference
		13.203
		16.918
		21.547
		8.338
		9.629
		4.437
		17.028
		-3.529
		10.295
		20.069
aveage x		11.79
standard deviation(s)		7.652

sample mean 'x̄=	11.794
sample size   n=	10.00
sample std deviation s=	7.65
std error 'sx=s/√n=	2.4199

for 95% CI; and 9 df, value of t=		2.262
margin of error E=t*std error    =		5.474
lower bound=sample mean-E =		6.320
Upper bound=sample mean+E =		17.267
from above 95% confidence interval for population mean =(6.32,17.27)

option A is correct