Most who run a One T Test in a statistical package like Minitab trust the P-Value to inform them whether to trust the Null Hypothesis or not. Just because “P is Low” doesn’t always mean that “the Ho must go”. Understanding the T Statistic will give you a more reliable determination. Below is an example of how to calculate the T Statistic for a One Sample T Test
A supplier of a part to a large organization claims that the mean (average) weight of a part is 90 grams. The organization took a small sample of 20 parts and found that the mean score is 84 grams and standard deviation is 11. Could this sample originate from a population of mean = 90 grams?
- Hypothesised Mean = 90
- N= 20 (# of Parts)
- DF= (N-1) = 19
- X-Bar (Average of the Samples) = 84
- S (Standard Deviation) = 11
The organization wants to test this at significance level of 0.05 (95% Confidence), i.e., it is willing to take only a 5 percent risk of being wrong when it says the sample is not from the population. Therefore:
- Null Hypothesis (H0): “True Population Mean Score is 90”
- Alternative Hypothesis (Ha): “True Population Mean Score is not 90”
- Alpha (Risk) is 0.05
Logically, the farther away the observed or measured sample mean is from the hypothesized mean, the lower the probability (i.e., the p-value) that the null hypothesis is true. However, what is far enough? In this example, the difference between the sample mean and the hypothesized population mean is 6. Is that difference big enough to reject H0? In order to answer the question, the sample mean needs to be standardized and the so-called t-statistics or t-value need to be calculated with this formula:
Don’t worry too much about the understanding the calculations because most statistical packages like Minitab, etc. will calculate the t-value for you.
Finally, this t-value must be compared with the critical value of t. You can find the Critical t Value on the following website: https://people.richland.edu/james/lecture/m170/tbl-t.html. Cross reference the ""Confidence Level with the "df" (Degrees of Freedom) to find the Critical t-value.
The critical t-value marks the threshold that – if it is exceeded – leads to the conclusion that the difference between the observed sample mean and the hypothesized population mean is large enough to reject H0. The critical t-value equals the value whose probability of occurrence is less or equal to 5 percent. From the t-distribution tables, one can find that the critical value of t is +/- 2.093.
Since the retrieved t-value of -2.44 is smaller than the critical value of -2.093, the null
hypothesis must be rejected (i.e., the sample mean is not from the hypothesized population) and the supplier’s claims must be questioned.
Has the P Value ever led you to the wrong conclusion? Have you compared the T Statistic to the Critical T Statistic and found that the P Value was giving “lying to you”?