Part 1. Student’s T-Test: One Sample T-Test

Author

Renato A. Folledo, Jr.

Introduction

A t-test is an inferential statistic that determine whether there is a significant difference between the mean of an observation with the mean of another group of observations. It establishes a null hypothesis that the two means are the same, and a research hypothesis that the 2 means are not the same.

There are 3 types of t-tests as shown below. For part 1, we will demonstrate the use of one-sample t-test to infer whether the mean bending of a set of observations has passed a certain strength set by the international standard.

Types of T-tests

One Sample T-test
T-test for Independent Samples
Paired (or dependent) T-test

1. One Sample T-Test

One-sample t-test compares the mean of observations to a known standard (or theoretical/hypothetical) mean (μ) value. For example, after developing a particle board using a new technique, we need to test whether its bending strength meets a certain standard. The table below show the bending strength for each of the 3 treatments. Now, let us use one-sample t-test whether the strength for each treatment meets a minimum standard strength of 30.

Table 1: Bending strength per treatment

treatment	strength
t1	39.3
t1	36.0
t1	31.2
t2	40.6
t2	40.8
t2	39.4
t3	35.3
t3	30.5
t3	39.0

Read the data and generate a box plot

ts <- data.table::fread("C:/transit/bending.csv")
boxplot(ts[,strength] ~ ts[,treatment], xlab="treatment", ylab = "bending strngth")

Figure 1: Box plot of bending strength per treatment

Figure 1 and Table 1 seem to show that all the observations from each treatment have passed the standard bending strength of 30. Now let us use one-sample t-test to verify it.

1. Treatment 1

Hypothesis:

H0: μ < 30 (The bending strength of Treatment 1 is not significantly greater than 30)

H1: μ > 30 (The bending strength of Treatment 1 is significantly greater than 30)

t.test(ts[treatment=="t1",strength], mu = 30, alternative = "greater")


    One Sample t-test

data:  ts[treatment == "t1", strength]
t = 2.3602, df = 2, p-value = 0.0711
alternative hypothesis: true mean is greater than 30
95 percent confidence interval:
 28.69719      Inf
sample estimates:
mean of x 
   35.493

Conclusion: Since the p-value of treatment 1 is 0.07110073, which is greater than 0.05, therefore, it failed to reject the null hypothesis that the bending strength of treatment 1 is not significantly greater than 30.

2. Treatment 2

Hypothesis:

H0: μ < 30 (The bending strength of Treatment 2 is not significantly greater than 30)

H1: μ > 30 (The bending strength of Treatment 2 is significantly greater than 30)

t.test(ts[treatment=="t2",strength], mu = 30, alternative = "greater")


    One Sample t-test

data:  ts[treatment == "t2", strength]
t = 23.677, df = 2, p-value = 0.0008896
alternative hypothesis: true mean is greater than 30
95 percent confidence interval:
 39.02271      Inf
sample estimates:
mean of x 
   40.292

Conclusion: Since the p-value of treatment 2 is 0.0008895528, which is less than 0.05. Therefore, the test rejects the null hypothesis. The bending strength of treatment 2 is significantly greater than 30.

3. Treatment 3

Hypothesis:

H0: μ < 30 (The bending strength of Treatment 3 is not significantly greater than 30)

H1: μ > 30 (The bending strength of Treatment 3 is significantly greater than 30)

t.test(ts[treatment=="t3",strength], mu = 30, alternative = "greater")


    One Sample t-test

data:  ts[treatment == "t3", strength]
t = 1.9961, df = 2, p-value = 0.09202
alternative hypothesis: true mean is greater than 30
95 percent confidence interval:
 27.71562      Inf
sample estimates:
mean of x 
 34.93567

Conclusion: Since the p-value of treatment 3 is 0.09201621, which is greater than 0.05, therefore, it failed to reject the null hypothesis that the bending strength of treatment 3 is not significantly greater than 30.

2. T-test for Independent Samples (to follow)

3. Paired T-Test (to follow)

Summary

There are 3 treatments tested. The 3 treatments vary depending on the ingredients used in making the cement bonded boards. Although the mean bending strength of the 3 treatments exceeded 30, the one sample t-test infers that only treatment 2 has a significantly stronger bending strength as set by the international standard. The rest of the treatments did not significantly meet the international standard of 30 units.

References:

https://jwoodscience.springeropen.com/articles/10.1186/s10086-021-02004-3

https://www.statology.org/one-sample-t-test-example-problems/