Introduction

In this report we compare aspirin concentrations in blood and also check if exercise reduces time taken for infants to walk.

Question 1: Aspirin Concentrations

asp_A <- c(15, 26, 13, 28, 17, 20, 7, 36, 12, 18)
asp_B <- c(13, 20, 10, 21, 17, 22, 5, 30, 7, 11)

(a) Hypotheses

Let \(\mu_d\) be the difference between the mean of the concentration of Aspirin A to Aspirin B.

\(H_0: \mu_d = 0\)

\(H_a: \mu_d \ne 0\)

(b) Paired t-test

t.test(asp_A, asp_B, paired = TRUE)
## 
##  Paired t-test
## 
## data:  asp_A and asp_B
## t = 3.6742, df = 9, p-value = 0.005121
## alternative hypothesis: true mean difference is not equal to 0
## 95 percent confidence interval:
##  1.383548 5.816452
## sample estimates:
## mean difference 
##             3.6

The p-value is 0.005121.

Since this is below 0.05, we reject \(H_0\) as the mean concentrations differ significantly.

(c) Two-sample t-test

t.test(asp_A, asp_B, paired = FALSE)
## 
##  Welch Two Sample t-test
## 
## data:  asp_A and asp_B
## t = 0.9802, df = 17.811, p-value = 0.3401
## alternative hypothesis: true difference in means is not equal to 0
## 95 percent confidence interval:
##  -4.12199 11.32199
## sample estimates:
## mean of x mean of y 
##      19.2      15.6

The two-sample p-value is 0.3401.

Since it exceeds 0.05we fail to reject \(H_0\).

Question 2: Infant Walking Times

active <- c(9.50, 10.00, 9.75, 9.75, 9.00, 13.00)
no_exercise <- c(11.50, 12.00, 13.25, 11.50, 13.00, 9.00)

(a) Hypotheses

Let \(\mu_A\) and \(\mu_N\) be the mean time taken to walk time for infants with active exercise and infants with no excercise respectively.

\(H_0: \mu_A = \mu_N\).

\(H_a: \mu_A < \mu_N\)

(b) Nonparametric Method

We might want to use a non-parametric method as each group has a very small number of infants, and hence can not be sure that the walking times follow a normal distribution.

(c) Mann–Whitney U Test

wilcox.test(active, no_exercise,
            alternative = "less",
            exact = FALSE,
            correct = TRUE)
## 
##  Wilcoxon rank sum test with continuity correction
## 
## data:  active and no_exercise
## W = 9, p-value = 0.08523
## alternative hypothesis: true location shift is less than 0

The test gives W = 9 and p = 0.08523.

Since p exceeds 0.05 we do not reject \(H_0\).

Hence there is insufficient evidence that active exercise leads to earlier walking in infants.