# 1-sample hypothesis test

Use 1-sample hypothesis tests to compare one sample with a target, for example, a 1-sample t-test.

## 1 proportion

Use a 1 proportion test to analyze the difference between an observed process proportion (defectives) and a specified value.

• Is the process proportion defective significantly different from a specified value such as a previously known defective proportion or a nominal specification?
When to Use Purpose
Pre-project Verify the process is producing output significantly different from expectations (in this case, that usually means a higher-than-expected defect rate), which validates the need for an improvement project.
Mid-project Test whether the proportion defective changed significantly when an input is controlled at a new setting or a previously uncontrolled setting is now controlled.
Mid-project Verify changes from the pre-project standard, throughout the course of making improvements.
End of project Verify the proportion defective from the controlled improved process is different from the pre-project proportion defective. Of course, this step assumes one of the goals of the project was to reduce the proportion defective.

### Data

Your data must be discrete Y at exactly two levels (also called binary or binomial data). You can enter the raw data into a single column in Minitab where each row represents one observation. Or, you can enter summarized data (the number of items sampled and the number of defectives observed) in the 1 Proportion dialog box.

### Guidelines

• Develop a sound data collection strategy to ensure your conclusions are based on truly representative data.
• Use Minitab’s power and sample size command to determine the sample size necessary to detect the smallest difference of interest with sufficient power.
• The data must fit a binomial distribution, which means the data must meet the following assumptions:
• Each test result has exactly two possible outcomes.
• The probability of a particular outcome is constant for all trials.
• The trials are independent of each other.
• A 1 proportion test can also be used to generate a confidence interval for a proportion. For example, if you make 2,518 units and 239 are defective, you can use this test to state with 95% confidence that the defect rate is between 8.37% and 10.70%.
• By default, Minitab uses the exact method to perform the test and calculate the confidence interval. Minitab also provides a normal approximation method; however, this method is not recommended because it is not as accurate as the exact method.

### How-to

1. Verify the measurement system for the Y data is adequate.
2. Develop a data-collection strategy (who should collect the data, in addition to where and when; how many data values to collect; the data’s precision how to record the data; and so on).
3. Collect process data and enter the values into a single column in a Minitab worksheet. When you enter raw data, you can use any coding scheme, but only if you use two values. By default, Minitab defines the higher numeric value, or the text value closer to the end of the alphabet, as the event (defect). For example, you can enter the data as 0 for good and 1 for defective or normal/problem. If necessary, you can change the value order Minitab uses to define defectives.
4. Enter the test proportion (a standard or benchmark proportion) to compare the process data against.
5. Determine your hypothesis. You are often trying to prove the alternative hypothesis (Ha) with the data. In a 1 proportion test, the alternative hypothesis tests whether the process proportion defective is greater than, less than, or not equal to the benchmark value. The null hypothesis (Ho) is the opposite of the alternative hypothesis.
6. You can also perform the test without the actual data if you know the number of items sampled and the number of defective items.

## 1-sample t

Use a 1-sample t-test to analyze the difference between an observed process mean and a specified value.

• Is the mean of the process significantly different from a specified value such as a previously known value of the process mean or a nominal specification?
When to Use Purpose
Pre-project Verify the process is producing output significantly different from expectations, which validates the need for an improvement project.
Mid-project Test whether the output changes significantly when an input is controlled at a new setting or a previously uncontrolled setting is now controlled.
Mid-project Verify changes from the pre-project standard throughout the course of making improvements.
End of project Verify the mean output from the controlled improved process is different from the pre-project mean. Of course, this step assumes that one of the goals of the project was to shift the process mean.

### Data

Your data must be continuous Y (output) values.

### Guidelines

• Develop a sound data collection strategy to ensure your conclusions are based on truly representative data.
• Use Minitab’s Power and Sample Size command to determine the sample size necessary to detect the smallest difference of interest with sufficient power.
• The data must be continuous and reasonably normal. A 1-sample t-test is very robust to violations of the normality assumption, especially if the sample size is large (n > 25).
• It is good practice to graph your data when you use a statistical test. For a 1-sample t-test, use normal probability plots to evaluate normality and histograms to identify outliers.
• If you have discrete numeric data from which you can obtain every equally spaced value and you have measured at least 10 possible values, you can evaluate these data as if they are continuous.

### How-to

1. Verify the measurement system for the Y data is adequate.
2. Develop a data collection strategy. For example, determine who should collect the data, where and when the data should be collected, how many data values are needed, the preciseness of the data, how to record the data, and so on.
3. Collect process data, and enter the values into a single column in a Minitab worksheet.
4. Enter the standard or benchmark to compare the process data against.
5. Determine your hypothesis. You are trying to prove the alternative hypothesis (Ha) with the data. In a 1-sample t-test, the alternative hypothesis states that the process mean is greater than, less than, or not equal to a benchmark value. The null hypothesis (Ho) is the opposite of the alternative hypothesis.
6. You can also perform the test without the raw data if you know the mean, standard deviation, and sample size.

## 1-sample Wilcoxon

Use a 1-sample Wilcoxon test to analyze the observed difference between a process median and a known value. This test is similar to the 1-sample t-test, and is used as an alternative test for cases in which the data are not reasonably normal.

• Is the median of the process significantly different from a known standard (for example, a previous known value of the process mean or a nominal specification)?
When to Use Purpose
Pre-project Verify that the process is producing output significantly different from expectations, which validates the need for an improvement project.
Mid-project Test whether the output changes significantly when an input is controlled at a new setting or a previously uncontrolled setting is now controlled.
Mid-project Verify changes from the pre-project standard, throughout the course of making improvements.
End of project Verify that the median output from the controlled improved process is different from the pre-project median. This step assumes one of the goals of the project was to shift the process median.

### Data

Your data must be a continuous value for Y (output).

### Guidelines

• Develop a sound data-collection strategy to ensure your conclusions are based on truly representative data.
• The 1-sample Wilcoxon test is from a category called nonparametric statistics. The test is meant to be an alternative to the parametric test (1-sample t-test) for cases where the normality assumption fails badly. This failure does not occur frequently because of the robustness of the parametric tests, so parametric tests are recommended for the majority of cases.
• The data must be continuous, and the distribution must be reasonably symmetric. The Wilcoxon test is less powerful than the t-test, as long as the data are reasonably normal. In fact, for skewed data, the Wilcoxon test is not a good alternative to the t-test, which is very robust for skewed data. A third alternative is the 1-sample sign test, which does not make any assumptions about the distribution of the data. However, the Wilcoxon test is more powerful than the sign test if the data are reasonably symmetric.
• For these reasons, you should use the:
• 1-sample t-test when the data are reasonably normal (or the sample size is large)
• 1-sample Wilcoxon test as an alternative to the 1-sample t-test as long as the data are reasonably symmetric
• 1-sample sign test as a last resort
• It is always good practice to graph your data whenever you use a statistical test. For the 1-sample Wilcoxon test, use a histogram or normal probability plot to evaluate for reasonable normality or reasonable symmetry. You can also use the histogram to identify outliers.
• If you have discrete numeric data from which you can obtain every equally spaced value, and you have measured at least 10 possible values, your data often are evaluated as though they are continuous.

### How-to

1. Verify the measurement system for the Y data is adequate.
2. Develop a data collection strategy (who should collect the data, as well as where and when; how many data values are needed; the preciseness of the data; how to record the data, and so on).
3. Collect process data, and enter the values into a single column in a Minitab worksheet.
4. Enter the standard or benchmark value to compare the process data against.
5. Determine your hypothesis. You are trying to prove the alternative hypothesis (Ha) with the data. In a 1-sample Wilcoxon test, the alternative hypothesis tests whether the process median is greater than less than, or not equal to a benchmark value. The null hypothesis (Ho) is the opposite of the alternative hypothesis.

## 1 variance test

Use a 1 variance test to analyze the difference between an observed process standard deviation (or variance) and a specified value.

• Is the variability of the process significantly different than a specified value (for example, a known standard or a previous known value of the process standard deviation)?
When to Use Purpose
Pre-project Verify the variability of the process is significantly different from expectations, validating the need for an improvement project.
Mid-project Test whether a significant change has occurred in the variability of the output when an input is controlled at a new setting or a previously uncontrolled setting is now controlled.
Mid-project Verify changes from the pre-project standard, throughout the course of making improvements.
End of project Verify the variability of the controlled improved process is different from the pre-project variability. Of course, this assumes that one of the goals of the project was to reduce the variability of the process.

### Data

Your data must be continuous Y (output) values.

### Guidelines

• Develop a sound data collection strategy to ensure that your conclusions are based on truly representative data.
• Use Minitab’s Power and Sample Size command to determine the sample size necessary to detect the smallest difference of interest with sufficient power.
• Minitab calculates results based on two methods: the standard and adjusted methods. The standard method is more powerful, but requires that the data are reasonably normal. The adjusted method is less powerful; therefore, it is used only for nonnormal data. If the data are reasonably normal, use the standard method. If the data are not reasonably normal, use the adjusted method.
• It is good practice to graph your data when you use a statistical test. For a 1 variance test, use the histogram or normal probability plots to evaluate normality. Use histograms to identify outliers.
• If you have discrete numeric data from which you can obtain every equally spaced value and you have measured at least 10 possible values, you can evaluate these data as if they are continuous.

### How-to

1. Verify the measurement system for the Y data is adequate.
2. Develop a data collection strategy (who should collect the data, as well as where and when; how many data values are needed; the preciseness of the data; how to record the data, and so on).
3. Collect process data, and enter the values into a single column in a Minitab worksheet.
4. Enter the standard or benchmark value of the standard deviation to compare the process data against. Although this is called a 1 variance test, Minitab tests the standard deviation (the square root of the variance) by default.
5. Determine your hypothesis. The alternative hypothesis (Ha) is often what you are trying to prove with the data. The alternative hypothesis for a 1 variance test can state that the process standard deviation is greater than, less than, or not equal to a benchmark value. The null hypothesis (Ho) is the opposite of the alternative hypothesis.
6. You can also perform the test without the actual data if you know the standard deviation and sample size.

## 1-sample sign

Use a 1-sample sign test to analyze the observed difference between a process median and a known value. This test is similar to a 1-sample t-test, and is used as an alternative for cases where the data are not reasonably normal.

• Is the median of the process significantly different from a known standard such as a previously known value of the process mean or a nominal specification?
When to Use Purpose
Pre-project Verify the process is producing output significantly different from expectations, validating the need for an improvement project.
Mid-project Test whether the output changes significantly when an input is controlled at a new setting or a previously uncontrolled setting is now controlled.
Mid-project Verify changes from the pre-project standard throughout the course of making improvements.
End of project Verify the median output from the controlled improved process is different from the pre-project median. Of course, this step assumes that one of the goals of the project was to shift the process median.

### Data

Your data must be a continuous value for Y (output).

### Guidelines

• Develop a sound data collection strategy to ensure your conclusions are based on truly representative data.
• The 1-sample sign test is from a category called nonparametric statistics. The test is meant to be an alternative to the parametric test (1-sample t-test) for cases where the normality assumption fails badly. This failure does not occur frequently because of the robustness of the parametric tests; therefore, the parametric tests are recommended for the majority of cases.
• The data must be continuous; however, the sign test does not have any requirements about the distribution.
• The 1-sample sign test is less powerful than the 1-sample t-test as long as the data are reasonably normal. Also, the 1-sample Wilcoxon test provides a third alternative for testing process location; however, it assumes that the distribution is symmetric (not necessarily normal). The Wilcoxon test is more powerful than the sign test if the data are reasonably symmetric.
• For these reasons, you should use the:
• 1-sample t-test when the data are reasonably normal (or the sample size is large)
• 1-sample Wilcoxon test as an alternative to the 1-sample t-test as long as the data are reasonably symmetric
• 1-sample sign test as a last resort
• It is always good practice to graph your data whenever you use a statistical test. For the 1-sample sign test, use a histogram or normal probability plot to evaluate for reasonable normality. You can also use the histogram to identify outliers.
• If you have discrete numeric data from which you can obtain every equally spaced value, and you have measured at least 10 possible values, your data often are evaluated as though they are continuous.

### How-to

1. Verify the measurement system for the Y data is adequate.
2. Develop a data collection strategy (who should collect the data, as well as where and when; how many data values are needed; the preciseness of the data; how to record the data, and so on).
3. Collect process data, and enter the values into a single column in a Minitab worksheet.
4. Enter the standard or benchmark value to compare the process data against.
5. Determine your hypothesis. You are trying to prove the alternative hypothesis (Ha) with the data. In a 1-sample sign test, the alternative hypothesis tests whether the process median is greater than less than, or not equal to a benchmark value. The null hypothesis (Ho) is the opposite of the alternative hypothesis.