, and jpg. Suppose a researcher is interested in the effectiveness of a new insu
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Question
, and jpg. Suppose a researcher is interested in the effectiveness of a new insulin drug at decreasing fasting glucose levels. In this particular study, the research team will use each participant as his own control, offering the participants the study drug either preceding or proceeding a control period involving best standard of care. Assume the following sample off participarits enrolled in the study represents an appropriate approximation to the normal distribution. 1. Fasting Glucose (mg/dL) With Best standard care 134 129 128 123 127 134 124 121 131 137 Fasting Glucose (mg/dL) With new drug Participant 113 114 102 106 109 101 104 116 124 0 10 Calculate the change in fasting glucose with the new drug and with the best standard care within participant (that is calculate the delta for each participant). (2 points) Calculate the mean and the standard deviation of the delta. (2 points) Conduct a paired-sample t-test to investigate the effectiveness at reducing the fasting glucose level with the new insulin drug. Show all hypothesis testing steps. (10 points) a) b) c) d) Calculate the 95% confidence interval. Show all your work. (4 points) e) Compare your results in c and d above? What do you conclude? (2 points)Explanation / Answer
Solution:
Part a
The table for the delta values (Di) is given as below:
Di = Xi - Yi
Fasting Glucose with new drug (Xi)
Fasting Glucose with best standard care (Yi)
Di
123
134
-11
113
129
-16
114
128
-14
102
123
-21
106
127
-21
109
134
-25
101
124
-23
104
121
-17
116
131
-15
124
137
-13
Part b
The mean and standard deviation for delta values or Di’s is given as below:
Mean = Di/n
Di
(Di - Dbar)^2
-11
43.56
-16
2.56
-14
12.96
-21
11.56
-21
11.56
-25
54.76
-23
29.16
-17
0.36
-15
6.76
-13
21.16
Di = -176
(Di - Dbar)^2 = 194.4
n = 10
Mean = Dbar = Di/n = -176/10 = -17.6
SD = Sd = sqrt[ (Di - Dbar)^2/(n – 1)] = sqrt[194.4/(10 – 1)] = sqrt[194/9] = sqrt(21.55556) = 4.642796
SD = 4.642796
Part c
Here, we have to conduct paired t test for the difference between population means.
The null and alternative hypothesis is given as below:
Null hypothesis: H0: There is no any statistically significant difference in Fasting Glucose with new drug and fasting glucose with best standard care.
Alternative hypothesis: Ha: The average fasting glucose with new drug is less than the average fasting glucose with best standard care.
H0: µd = 0 versus Ha: µd < 0
This is lower tailed test.
Test statistic formula is given as below:
t = Dbar / [Sd/sqrt(n)]
We are given
Dbar = -17.6
Sd = 4.642796
n = 10
df = n – 1 = 10 – 1 = 9
We assume level of significance as = 0.05.
Test statistic is calculated as below:
t = -17.6/(4.642796/sqrt(10))
t = -11.9876
Lower critical value = -1.8331 (by using t-table)
P-value = 0.0000 (by using t-table)
= 0.05
P-value < = 0.05
So, we reject the null hypothesis that there is no any statistically significant difference in Fasting Glucose with new drug and fasting glucose with best standard care.
There is sufficient evidence to conclude that the average fasting glucose with new drug is less than the average fasting glucose with best standard care.
Part d
Confidence interval = Dbar -/+ t*Sd/sqrt(n)
We are given
Dbar = -17.6
Sd = 4.642796
n = 10
df = n – 1 = 10 – 1 = 9
Confidence level = 95%
Critical value t = 2.2622
(by using t-table)
Confidence interval = -17.6 -/+ 2.2622*4.642796/sqrt(10)
Confidence interval = -17.6 -/+ 2.2622* 1.4697
Confidence interval = -17.6 -/+ 3.32475534
Lower limit = -17.6 - 3.32475534 = -20.92475534
Upper limit = -17.6 + 3.32475534 = -14.27524466
Part e
In part c we reject the null hypothesis and conclude that the average fasting glucose with new drug is less than the average fasting glucose with best standard care.
Also, in part d, we find confidence interval which only contains negative values which means delta is negative or fasting glucose level is decreasing for new insulin drug.
Fasting Glucose with new drug (Xi)
Fasting Glucose with best standard care (Yi)
Di
123
134
-11
113
129
-16
114
128
-14
102
123
-21
106
127
-21
109
134
-25
101
124
-23
104
121
-17
116
131
-15
124
137
-13
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