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Student's t-test
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Using the t-distribution to compare a sample mean to a population mean with unknown variance.
Want a deeper conceptual understanding? Try our interactive lesson!
1 tailed and 2 tailed T-test hypotheses
SL AI 4.11
Given a null hypothesis H0:μ=μ0, we can have any of the following alternative hypotheses
H1:μ<μ0,H1:μ>μ0,H1:μ=μ0.
The first two alternative hypotheses are called one-tailed since we only care how far the sample mean, xˉ, is from μ0 in one direction. The last hypothesis is two-tailed because we care how far xˉ is from μ0 regardless of direction.
T-test for mean μ (1-sample)
SL AI 4.11
We can perform a t-test for a single sample against a known mean by on a calculator:
Enter the sample data into a list.
Navigate to T-Test on a calculator.
Select "DATA" and enter the name of the list where sample is stored.
Select the tail type depending on what our alternative hypothesis is (μ0 is the population mean):
=μ0 for a change in mean
<μ0 for a decrease in mean
>μ0 for an increase in mean
Hit calculate, and interpret the p-value as usual.
2-sample T-Test
SL AI 4.11
To compare the means of two samples using a T-test, we use a calculator:
Enter each sample in its own list.
Navigate to 2-SampTTest.
Select "Data", then enter the names of the lists containing the samples.
Select the tail type depending on what our alternative hypothesis is:
μ1=μ2 for different means
<μ2 for first list mean smaller than second
>μ2 for first list mean greater than second
Set "Pooled" to true.
The calculator reports the t-value and p-value, which we interpret as usual.
Paired tests for the mean
AHL AI 4.18
We say that data is paired when each value in one row is tied to the value in the next row. An example of this is before and after scores for a group of students.
Instead of comparing means using a two-sample test, we instead calculate the difference between the rows for each column, and then do a one-sample test with μ0=0.
Two important notes:
H0:d=0 and H1 can be d=0,d<0 or d>0.
The assumption being made is that the differences are normally distributed, not the original values.
Testing for population correlation: H₀ : ρ = 0 vs H₁ : ρ ≠ 0
AHL AI 4.18
We can test the correlation between two normally distributed populations.
Enter the samples into L1 and L2
Navigate to LinRegTTest.
Select ρ=0,<0 or >0
The calculator returns the p-value (not to be confused with ρ), as well as the coefficients y=a+bx.
Nice work completing Student's t-test, here's a quick recap of what we covered:
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