Homework SoursePlacebo is a harmless pill given to participants in medical
Placebo is a harmless pill given to participants in medical research. It doesn’t have any effect, but a lot of participants will show improvement in medical condition after taking the pill. This is called a placebo effect. A study had 12 participants with similar medical condition (High Blood Pressure) and 6 of them were randomly selected to take placebo pills, and the other 6 took nothing. After 20 minutes, their blood pressure were measured as below. Placebo 135 141 130 129 132 129 Nothing 143 145 138 142 138 137 (1) (9pts)What can you conclude about the average blood pressure for placebo group (95% confidence interval)? (2) (9pts)What can you conclude about the average blood pressure for those took nothing (95% confidence interval)? (3) (3pts)Are there any overlap between those two confidence intervals?
Solution
We have given that placebo is a harmless pill given to participants in medical research. It doesn’t have any effect, but a lot of participants will show improvement in medical condition after taking the pill. This is called a placebo effect.
A study had 12 participants with similar medical condition (High Blood Pressure) and 6 of them were randomly selected to take placebo pills, and the other 6 took nothing. After 20 minutes, their blood pressure were measured as below
Placebo 135 141 130 129 132 129
Nothing 143 145 138 142 138 137
a) What can you conclude about the average blood pressure for placebo group (95% confidence interval)?
We have to calculate confidence interval for average blood pressure for placebo.
The 95% confidence interval for mean is,
Xbar - E < mu < Xbar + E
Where Xbar is the sample mean for placebo.
mu is the population mean for placebo.
E is the margin of error.
E = tc * s / sqrt(n)
tc is the 95% critical value for t-distribution degrees of freedom are n - 1.
s is the sample standard deviation for placebo group.
Placebo 135 141 130 129 132 129
n = 6
Xbar = sum of ovservations/ total number of observations = 132.6667
s = sum of (X - Xbar)2 / n-1 = 4.6762
By using t-table the critical value or tc is 2.015
E = 2.015 * 4.6762 / sqrt(6) = 3.8467
Lower limit = Xbar - E = 132.6667 - 3.8467 = 128.82
Upper limit = Xbar + E = 132.6667 + 3.8467 = 136.5134
The 95% confidence interval for placebo group will be (128.82 , 136.5134).
b) What can you conclude about the average blood pressure for those took nothing (95% confidence interval)?
We have to calculate confidence interval for average blood pressure for those took nothing.
The 95% confidence interval for mean is,
Xbar - E < mu < Xbar + E
Where Xbar is the sample mean for those took nothing.
mu is the population mean for those took nothing.
E is the margin of error.
E = tc * s / sqrt(n)
tc is the 95% critical value for t-distribution degrees of freedom are n - 1.
s is the sample standard deviation for those took nothing.
Nothing 143 145 138 142 138 137
n = 6
Xbar = sum of ovservations/ total number of observations = 140.5
s = sum of (X - Xbar)2 / n-1 = 3.2711
By using t-table the critical value or tc is 2.015
E = 2.015 * 3.2711 / sqrt(6) = 2.6909
Lower limit = Xbar - E = 140.5 - 2.6909 = 137.8091
Upper limit = Xbar + E = 140.5 - 2.6909 = 143.1909
The 95% confidence interval for those took nothing will be (137.8091 , 143.1909).
Length of confidence interval for placebo = 136.5134 - 128.82 = 7.6934
Length of confidence interval for those took nothing = 143.1909 - 137.8091 = 5.3818
The confidence limits for placebo are smaller than that of those took nothing.
