Student's T-test
This lesson covers:
- How the student's t-test can be used to compare means between two datasets
Comparing means using student’s t-test
The student’s t-test is a statistical test used to determine if there is a significant difference between the mean values of a particular variable across two populations.
The conditions for using student's t-test are:
- The data must be continuous and normally distributed.
- The variances of the populations should be equal.
- The samples must be independent of each other.
Note: variance is the standard deviation squared. You don't need to learn the formula, but it is:
variance =n−1Σ(x−xˉ)2
Conducting a t-test
Step 1 - State the null hypothesis.
This assumes there is no significant difference between the means of the datasets.
Step 2 - Calculate the t statistic using the t-test formula (this will be provided in exams).
t=√(n1σ12)+(n2σ22)(xˉ1−xˉ2)
Where:
- xˉ1= mean of the first dataset
- xˉ2= mean of the second dataset
- σ1= standard deviation of the first dataset
- σ2= standard deviation of the second dataset
- n1= sample size of the first dataset
- n2= sample size of the second dataset
Step 3 - Calculate the degrees of freedom.
degrees of freedom (df) in a t−test = n1+n2−2
Step 4 - Compare the calculated t statistic against a critical value.
The critical value is determined by the degrees of freedom and the chosen significance level (usually p = 0.05). You will be provided with a table of critical values in an exam.
If the t statistic is greater than the critical value:
- Reject the null hypothesis.
- This suggests that the means are significantly different.
If the t statistic is less than the critical value:
- Accept the null hypothesis.
- This suggests that there is no significant difference between the means, and any difference is just due to chance.
Worked example - Comparison of turtle egg hatch rates using student's t-test
Given the following data, use the student's t-test to determine if there is a significant difference between the mean number of turtle eggs hatched in high and low temperature conditions.
| Dataset | Mean (number of eggs) | SD (number of eggs) | Sample size (n) |
|---|---|---|---|
| High temperature | 80 | 5 | 15 |
| Low temperature | 70 | 5 | 15 |
Step 1: Equation
t=√(n1σ12)+(n2σ22)(xˉ1−xˉ2)
Step 2: Substitution and correct evaluation
t=√(1552)+(1552)(80−70)
t=√(1525)+(1525)(10)
t=√1.66...+1.66...10
t=√3.33...10
t=1.82...10
t=5.48 (to 3 s.f.)
Step 3: Calculate degrees of freedom (df)
df in a t−test = n1+n2−2
df=(15+15)−2=28
Step 4: Determine significance
with a t value of 5.48 and df of 28, we compare this to the critical t value at a significance level of 0.05
the critical value for the t-test statistic at p = 0.05 and 28 degrees of freedom is 2.05
as our calculated t value (5.48) is greater than this critical value, we reject the null hypothesis and conclude that there is a significant difference between the hatch rates of turtle eggs in high and low temperatures