Question

In: Statistics and Probability

Recall in our discussion of the normal distribution the research study that examined the blood vitamin...

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 19 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 40.540 29.136
2 45.629 49.292
3 25.171 28.019
4 29.004 23.442
5 42.862 36.110
6 39.080 33.379
7 29.028 27.259
8 34.390 32.395
9 30.548 40.320
10 42.900 25.647
11 35.518 34.852
12 40.099 34.289
13 37.542 32.376
14 40.033 32.477
15 34.071 30.036
16 35.383 31.935
17 36.790 33.801
18 37.895 37.801
19 48.683 28.390

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. -8.95 to -1.03 ng/mL

b. -8.99 to -1.27 ng/mL

c. -7.75 to -1.11 ng/mL

d. -8.53 to -1.00 ng/mL

Solutions

Expert Solution

for 95% CI; and 18 degree of freedom, value of t= 2.101
therefore confidence interval=sample mean -/+ t*std error
margin of errror          =t*std error=             3.321
lower confidence limit                     = -7.75
upper confidence limit                    = -1.11

option C is correct :  -7.75 to -1.11 ng/mL


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