One Sample t test for mean
In this tutorial we will explain the six steps approach used in hypothesis testing to test hypothesis about the population mean when the population standard deviation is unknown.
One sample t test for mean
Let X1,X2,⋯,Xn be a random sample from a normal population with mean μ and unknown variance σ2. Let ¯x=1n∑xi be the sample mean and s2=1n−1∑(xi−¯x)2 be the sample variance.
Assumptions
a. The population from which, the sample drawn is assumed as Normal distribution.
b. The population variance σ2 is unknown.
Step by Step Procedure
We wish to test the null hypothesis H0:μ=μ0, where μ0 is the specified value of the population mean.
The standard error of mean is
SE(¯x)=σ√n=s√n
Step 1 State the hypothesis testing problem
The hypothesis testing problem can be structured in any one of the three situations as follows:
Situation | Hypothesis Testing Problem |
---|---|
Situation A | H0:μ=μ0 against Ha:μ<μ0 (Left-tailed) |
Situation B | H0:μ=μ0 against Ha:μ>μ0 (Right-tailed) |
Situation C | H0:μ=μ0 against Ha:μ≠μ0 (Two-tailed) |
Step 2 Define the test statistic
The test statistic for testing above hypothesis is
t=¯x−μSE(¯x)=¯x−μ0s/√n
The test statistic t follows Students’ t distribution with n−1 degrees of freedom.
Step 3 Specify the level of significance α.
Step 4 Determine the critical values
For the specified value of α determine the critical region depending upon the alternative hypothesis.
- left-tailed alternative hypothesis: Find the t-critical value using
P(t<−tα,n−1)=α.
- right-tailed alternative hypothesis: tα.
P(t>tα,n−1)=α.
- two-tailed alternative hypothesis: tα/2.
P(|t|>tα/2,n−1)=α.
Step 5 Computation
Compute the test statistic under the null hypothesis H0 using equation
tobs=¯x−μ0s/√n
Step 6 Decision (Traditional Approach)
Traditional approach is based on the critical value.
- For left-tailed alternative hypothesis: Reject H0 if
tobs≤−tα,n−1
. - right-tailed alternative hypothesis: Reject H0 if
tobs≥tα,n−1
. - two-tailed alternative hypothesis: Reject H0 if
|tobs|≥tα/2,n−1
.
OR
Step 6 Decision (p-value Approach)
It is based on the p-value.
Alternative Hypothesis | Type of Hypothesis | p-value |
---|---|---|
Ha:μ<μ0 | Left-tailed | p-value =P(t≤tobs) |
Ha:μ>μ0 | Right-tailed | p-value =P(t≥tobs) |
Ha:μ≠μ0 | Two-tailed | p-value =2P(t≥tobs) |
If p-value is less than α, then reject the null hypothesis H0 at α level of significance, otherwise fail to reject H0 at α level of significance.