## Because “P is Low” doesn’t mean “the Ho must go”; One Sample T (T Statistic Explanation)

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

Example:

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”?

## Comments

Let's try it this way. You state "Just because “P is Low” doesn't always mean that “the Ho must go”. " However, your example is a case when the p value is low and you conclude Ho must go.

You need an example where "P is low" and you conclude that "Ho must NOT go." For the one sample two-sided t-test with alpha risk = 5%, you need a case where p < 0.05 and |t-value| < t-critical.

You state this in your reply: “I have seen on multiple occasions where the p-value pointed you towards one determination and the t-value was the opposite.”

Show us one of those case.

The p-value should only support the t-value. I have seen on multiple occasions where the p-value pointed you towards one determination and the t-value was the opposite. In the case of this article the p-value supported the t-value.

You have to be careful of having a representative sample size Dr. Mikel Harry in comments to this article explains this well (http://linkd.in/1j01ELg): "We must remember, there are 5 key parameters that govern statistical significance – alpha risk, beta risk, delta, variance and sample size. Hold any 3 constant and the 4th will vary as the 5th varies (and conversely). For example, if alpha, delta and the variance are fixed, then beta will be reduced as sample size increases. Understanding each of these relationships is the key to grasping the idea of hypothesis testing."

Since the retrieved t-value of -2.44 is smaller than the critical value of -2.093, the null hypothesis must be rejected.

So is the conclusion from p-value!

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