8.1.1 Performing a One-Tailed One-Sample T-Test

For a one-tailed one-sample t-test, we use the T.INV function in Excel.

The syntax is: =T.INV(probability, degrees_freedom)

Figure 24 T.INV Function

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The image shows an Excel function, T.INV, which calculates the inverse of the Student’s t-distribution. It requires two inputs: the probability associated with the t-distribution and the degrees of freedom (deg_freedom), which defines the specific t-distribution.

Figure 24 T.INV Function

Where:

• probability: This value represents the cumulative probability for which you want to find the corresponding t-value. It determines the boundary of the t-distribution.
• degrees_freedom: This equals N-1, where N is the sample size. It defines the specific shape of the t-distribution.

Let us explain the "probability" parameter further, as it relates to the boundary of the t-distribution:

• The probability input determines where on the t-distribution you want to find the t-value.

• For a one-tailed test with a 95% confidence level:
  • Use 0.05 for an upper-tailed test
  • Use 0.95 for a lower-tailed test

• The resulting t-value represents the boundary between the “rejection” and the “fail to reject” region of the hypothesis test.

For example, if we want to conduct a one-tailed one-sample t-test with a significance level of 0.05 and a sample size of 11 (giving 10 degrees of freedom):

The lower tail refers to the left side of the t-distribution.

• Formula: To find the critical value for the lower tail, use: =T.INV(0.05, 10)  , which return -1.812

The upper tail refers to the right side of the t-distribution.

• Formula: To find the critical value for the upper tail, use: =T.INV(0.95, 10), which return 1.812

• This boundary is crucial because:
  • In an upper-tailed test, if your calculated t-statistic is greater than this boundary, you reject the null hypothesis.
  • In a lower-tailed test, if your calculated t-statistic is less than this boundary, you reject the null hypothesis.

• The shape of the t-distribution, and thus these boundaries, changes with the degrees of freedom. As degrees of freedom increase, the t-distribution approaches the normal distribution.