### Table of contents

### When the “P” is Low, the “Ho” Must Go!

Most people who run one sample t-test in a statistical package like Minitab trust the P-Value to inform them whether to trust the Null HypothesisHypothesis testing definition A statistical hypothesis test ... Learn More....

Just because the “P is Low,” it doesn’t always mean that “the Ho Must Go.” Understanding the issues and inputs related to the One Sample T-test will give you a more reliable determination.

Below is an example of a One Sample T Test and the most common issue that is challenged:

### Example One Sample T Test

A supplier of a part to a large organization claims that the average(mean) 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 the standard deviationCumulative probability of a normal distribution with expecte... Learn More... is 11. Could this sample originate from a population of mean = 90 grams?

- Hypothesised Mean = 90
- N Sample size (# of Parts) = 20
- X-BarWhat is an X-Bar? In statistics, an X-bar (also written as x... Learn More... (AverageA synonym for “mean”: the sum of a set of values divided... Learn More... of the Samples) = 84
- S (Standard Deviation) = 11
- Delta or Change = 6

The organization wants to test this at a significance levelStatistics level A statistics level is the value of input in... Learn More... 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 HypothesisA statistical hypothesis is a hypothesis that is testabl... Learn More... (H0): “True Population MeanA synonym for “mean”: the sum of a set of values divided... Learn More... Score = 90”
- Alternative Hypothesis (Ha)An alternative hypothesis (Ha) states that there is a s... Learn More...: “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 (Delta) between the Sample Mean and the Hypothesized Population Mean is 6. Is that difference big enough to reject H_{0} based on the test results and information provided?

First, the sample sizeThe sample size is an important feature of any empirical stu... Learn More... needs to be large enough to detect the difference.

In this case, the required sample size to be able to detect a true difference between populations is 38 (use the One Sample T-test Power and sample sizePower and sample size The power and sample size estimates ar... Learn More... calculator in Minitab).

With a sample size of 20, if the difference is less than 8.4 the test is not definitive or sensitive enough to detect a true shift. If you use the supplied data in the summarized T-test you will get a low P value – which may be challenged and questioned by someone who is familiar with Hypothesis TestingHypothesis testing definition A statistical hypothesis test ... Learn More....

Don’t worry too much about understanding the calculations because most statistical packages, like Minitab, will calculate Sample Sizes, T-Test Statistics, and P-Value for you.

If interested, you can compare the T-Value generated 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 LevelMost of your process data is not possible to collect. You w... Learn More... 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-DistributionUsed for determining the confidence interval for means or fo... Learn More... 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 may be questioned. However, the base issue related to sample size needs to be addressed first.

Has the P-Value ever led you to the wrong conclusion? Was your sample size sufficient to make the correct determination?

**Share your thoughts in the comments below!**