# One-Sample Test

## Contents

### Overview

The one-sample test is a statistical procedure used to determine whether the mean of a given sample of data is statistically different from the mean of the population (or an hypothesized mean). A one-sample t-test is generally performed when the population standard deviation is unknown, otherwise a one-sample z-test is used.

Any Hypothesis test in statistics requires the definition of a null hypothesis and an alternate hypothesis. The goal of the test is to aid the analyst in deciding which one of the two hypothesis is true. The table below shows a common set of choices for the null and alternate hypothesis for a one-sample test:

Null hypothesis Alternate hypothesis
Mean = Hypothesized Mean Mean not equal to the Hypothesized Mean
Mean = Hypothesized Mean Mean < Hypothesized Mean
Mean = Hypothesized Mean Mean > Hypothesized Mean

Before running a statistical test, the analyst must choose a significance level, or the probability of rejecting the null hypothesis given that it were true (i.e. the probability of making a wrong decision). A significance level of 0.05 is usually adopted but a different value may be used depending on the field of the study. If the p-value obtained at the end of the test is less than the selected significance level, the Null Hypothesis should be rejected in favour of the Alternate Hypothesis.

More details on the One-Sample t- and z-test can be found here:

### Practical Example

Suppose a manufacturer wants to determine whether the cakes produced by a new manufacturing process weigh 3.0 Kg on average. He randomly selects 50 cakes from the 1000 produced in his factory in a day. A One-Sample t-Test can be performed with following hypothesis:

 Null hypothesis Mean = Hypothesized Mean Mean not equal to the Hypothesized Mean

The 50 cakes selected have a mean of 2.9 and a standard deviation of 0.2. The results of a one-sample t-test indicates a p-value of 0.001. Being the p-value less than the adopted significance level of 0.05, the manufacturer should reject the null hypothesis and conclude that the difference between the sample mean and the hypothesized mean is statistically significant.

### Within Grapheme

To perform a One Sample Test in Grapheme, click on Create New Analysis from the toolbar of the Statistical Analysis View. Assign a name to the panel and select “One Sample Test” from the list. Click on Next.

Select the Source Table from the ones available in the Tables view, select the view of the table and the column containing the sample you want to analyse. Click on Next.

Check the condition (or alternate hypothesis) you want to verify and define the Test Settings as follow

• Select a T-Test if the standard deviation of the population is unknown. Once selected insert the Population Mean value.
• Select a Z-Test if the standard deviation of the population is unknown. Once selected insert the Population Mean and standard deviation values.

If you want Grapheme to evaluate the confidence interval range, check the setting Compute Confidence Interval and insert the probability value to be used for its evaluation. Click on Finish.

##### Remarks
• All the data available in the panel are updated on the fly, so that any change in the table values, is immediately reflected by the panel tables and charts. Automatic update can be temporary suspended, by clicking on the lock button in the main panel toolbar.
• All the data contained in each table, can be copied to the clipboard for further reporting by clicking on the button available in the toolbar.
• Each chart can be exported as Image by clicking on the “Save as Image” button available in the toolbar