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=1nxi be the sample mean and s2=1n1(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)=σn=sn

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:μμ0 (Two-tailed)

Step 2 Define the test statistic

The test statistic for testing above hypothesis is t=¯xμSE(¯x)=¯xμ0s/n

The test statistic t follows Students’ t distribution with n1 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,n1)=α.
  • right-tailed alternative hypothesis: tα. P(t>tα,n1)=α.
  • two-tailed alternative hypothesis: tα/2. P(|t|>tα/2,n1)=α.

Step 5 Computation

Compute the test statistic under the null hypothesis H0 using equation tobs=¯xμ0s/n

Step 6 Decision (Traditional Approach)

Traditional approach is based on the critical value.

  • For left-tailed alternative hypothesis: Reject H0 if tobstα,n1.
  • right-tailed alternative hypothesis: Reject H0 if tobstα,n1.
  • two-tailed alternative hypothesis: Reject H0 if |tobs|tα/2,n1.

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(ttobs)
Ha:μ>μ0 Right-tailed p-value =P(ttobs)
Ha:μμ0 Two-tailed p-value =2P(ttobs)

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.

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