Question

A double-blind randomised experiment investigated whether consumption of oligofructose would stimulate calcium absorption in healthy human males aged 14 to 16 years. Two sets of pills were prepared, identical in appearance and packaging. One set contained oligofructose and the other set contained ordinary sucrose (i.e., a placebo). The experiment ran in two rounds, with the second round starting 3 weeks after the first ended. Subjects were randomly assigned to either receive oligofructose in the first round and the placebo in the second, or the placebo in the first round and oligofructose in the second. In each round, subjects took pills for nine days and had their calcium absorption measured on the last day. Calcium absorption was measured as a percentage of calcium intake. The calcium absorption data are available in two data files, Oligo_paired.csv and Oligo_indep.csv. The measurements in each file are the same, but only one of these files is appropriate to the problem. In Oligo_paired.csv, the data columns are headed “Placebo” and “Oligofructose” to indicate which pill was being taken when calcium absorption was measured. Each row contains calcium absorption measurements from a single subject. In Oligo_indep.csv, the columns are headed “CaAbsorb” and “Pill”. CaAbsorb contains calcium absorption measurements, and “Pill” indicates whether the pill being taken was a placebo or oligofructose.

Placebo	Oligofructose
68.4	72	-3.6
66.6	85.1	-18.5
57.4	56.5	0.9
51.5	59.4	-7.9
49	79.7	-30.7
46.6	53.8	-7.2
44.2	60.3	-16.1
42.9	61.6	-18.7
37.2	76.6	-39.4
34.1	62.7	-28.6
24.6	64	-39.4

CaAbsorb	Pill
68.4	Placebo
66.6	Placebo
57.4	Placebo
51.5	Placebo
49	Placebo
46.6	Placebo
44.2	Placebo
42.9	Placebo
37.2	Placebo
34.1	Placebo
24.6	Placebo
72	Oligofructose
85.1	Oligofructose
56.5	Oligofructose
59.4	Oligofructose
79.7	Oligofructose
53.8	Oligofructose
60.3	Oligofructose
61.6	Oligofructose
76.6	Oligofructose
62.7	Oligofructose
64	Oligofructose

(b) Provide a graph of the data distribution appropriate to your choice of analysis.

How do i create the graph in R?

c) Formally test whether oligofructose stimulates calcium absorption in healthy males aged 14 to 16 years. You do not need to revisit the decision made in part (a), but your report should be consistent with that decision, and should include all relevant elements of a hypothesis test. Assume α = .05.

What is the null and alternative hypothesis?

Answer 1

(b)

Plotting the differences we obtain as follows.

The R-code to obtain the above histogram is as follows.

x<-c(-3.6,-18.5,0.9,-7.9,-30.7,-7.2,-16.1,-18.7,-39.4,-28.6,-39.4)
hist(x,main="Histogram of differences",xlab="Difference",border="blue",col="green", xlim=c(-40,10),ylim=c(0,4),las=1,breaks=8)

Note-The R-code I used is given above. However, you may remove some parts like 'main', 'xlab', 'border', 'col', 'xlim', 'ylim', 'las' as well as 'breaks'. Merely using the R-code as below will give a histogram.

x<-c(-3.6,-18.5,0.9,-7.9,-30.7,-7.2,-16.1,-18.7,-39.4,-28.6,-39.4)
hist(x)

We observe that the data is approximately normally distributed.

So, we can perform paired t-test.

(c)

Suppose, random variables X and Y denote calcium absorption using Placebo and Oligofructose respectively. Also, random variable D(=X-Y) denotes difference in calcium absorption.

We have to test for null hypothesis

against the alternative hypothesis

Our test statistics is given by

Here,

Number of pairs of observation

Degrees of freedom

[Using R-code 'pt(-4.498862,10)']

Level of significance

We reject our null hypothesis if

Here, we observe that

So, we reject our null hypothesis.

Hence, based on the given data we can conclude that there is significant evidence that Oligofructose stimulates calcium absorption.