One sample t-test

To undertake hypothesis testing it is necessary to go through these following stages:

1. Decide on the null hypothesis, H0.
2. Decide on the alternate hypothesis, H1.
3. Decide on the significance level.
4. Calculate the appropriate test statistic, using the sample data.
5. Calculate the p-value.
6. Compare the p-value with the significance level, and decide whether to reject or accept the null hypothesis, H0.

The null hypothesis is a statement about the population value that will be tested - “no change” statement. The alternative hypothesis is the hypothesis that includes all population values not covered by the null hypothesis - “some change” statement

There are several different types of hypothesis tests. Specifically, this post concentrates on the one sample t-test.

What is a one sample t test?
The one sample t-test is used to test the null hypothesis that the mean of the population from which the data sample is drawn is equal to a hypothesized value.

Hypotheses:

• Null: There is no significant difference between the sample mean and the population mean.
• Alternate: There is a significant difference between the sample mean and the population mean.

Assumptions
Before running a one sample t-test you need to establish if the data violates any assumptions by using these tests. Generally, the test assumes that the sample values are independent and are all identically normally distributed (same mean and variance). If your data do not come from a normal distribution:

• transform the values to make the distribution more normal
• use a nonparametric test instead, for example, a one-sample sign test. This is potentially the best option if the distribution is highly skewed
• use the t test anyway, knowing that the t test still works well if the underlying distribution
  is symmetric, unimodal, and continuous.

A fuller description of the possible alternatives can be found here.

One of the advantages of the t-test is that it can be applied to a relatively small number of cases. It was specifically designed to evaluate statistical differences for samples of 30 or less. However, although the assumption of normality is not too important with large samples, it is important with small sample sizes, for example less than 10.

You also need to be aware that statistically significant does not mean the same as practically significant.

Minitab
To run a one sample t-test for a hypothesis concerning a single mean in Minitab:

Stats -> Basic Statistics -> 1 sample t -> Variables (enter data columns)-> Check “Test of Mean” (enter value in the box).

Minitab will respond with Test of mu = (value you entered) vs mu not = (value you entered). Below this will be printed the N, mean, standard deviation, standard error, T statistic, and P value.

This link takes you to the Minitab training material for a one-sample t-test. Alternatively, this or this provide a nice summary. If you don’t have Minitab use this site.

P-value
P is the probability that the mean for the data equals the value you were comparing it to. The Prism site has a great summary about how to think about results from the one-sample t test.

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