Student’s T distribution leverages the T distribution and is used for finding confidence intervals for the population mean when the sample size is less than 30 and the population standard deviation is unknown.
What is a One Sample T Hypothesis Test?
The one sample t test allows to compares the (small) population mean to some hypothesized value or one sample mean to determine if they are significantly different.
For example, if we know the average weight of chickens in a farm is 3lb, and wish to compare average weight of sample black hens to the population mean value.
If you need to evaluate something with a population greater than 30, use the Z distribution .
Student t-test would be used to compare two population means using samples from each.
Also used to test hypotheses about population means based on sample data.
One Sample T Test Hypothesis
Null hypothesis (H0): The difference between population mean and the hypothesized value is equal to zero
Alternative hypothesis (H1):
- The population mean is not equal to hypothesized value (two-tailed)
- The population mean is greater than hypothesized value (upper-tailed)
- The population mean is less than hypothesized value (lower-tailed)
Right-tailed or upper-tailed test
Right tailed test is also called the upper tail test. A hypothesis test is performed if the population parameter is suspected to be greater than the assumed parameter of the null hypothesis.
- H0: The sampling mean (x̅) is less than are equal to µ
- H1: The sampling mean (x̅) is greater than µ.
Left-tailed test or lower-tailed test
Left-tailed test is also known as a lower tail test. A hypothesis test is performed if the population parameter is suspected to be less than the assumed parameter of the null hypothesis.
- H0: The sampling mean (x̅) is greater than are equal to µ
- H1: The sampling mean (x̅) is less than µ.
Assumptions of One Sample T Hypothesis Test
- Data is continuous and quantitative at the scale level (in other words data in ratio or interval)
- The sample should be randomly selected from the population
- Samples are independent to each other
- Data should follow normal probability distribution
- Assumes it don’t have extreme outliers in the dependent variable
When Would You Use a One Sample T Hypothesis Test?
One sample t test is a type of parametric test because the assumption is samples are randomly distributed. It tests whether the sample mean is significantly different than a population mean when the standard deviation of the population is unknown. Hence t test is used when the population standard deviation is unknown and the sample size is below 30, otherwise use Z-test (for known variance)
Steps to Calculate One Sample T Hypothesis Test
- State the claim of the test and determine the null hypothesis and alternative hypothesis
- Determine the level of significance
- Calculate degrees of freedom
- Find the critical value
- Calculate the test statistics
- x̅ is observed sample mean
- μ0 is population mean
- s is sample standard deviation
- n is the number of the observations in the sample
- Make a decision, the null hypothesis will be rejected if the test statistic is less than or equal to the critical value
- Finally, Interpret the decision in the context of the original claim.
Example of a One Sample T Hypothesis Test in a DMAIC Project
One Sample T test mostly performed in Analyze phase of DMAIC to check the significant difference between the population mean and the sample means, while paired t-test can be performed in Measure phase to review before and after process improvement (see below example for more details).
According to American health association the average blood pressure of a pregnant women is 120 mm Hg. Collected 15 random samples from pregnant women to check the sample blood pressure is different from accepted standard blood pressure.
- Null Hypothesis: No difference between sample data and population blood pressure (H0: μ=120)
- Alternative Hypothesis: There is a difference between sample data and population blood pressure (H1: μ≠120)
Significance level: α=0.05
Degrees of freedom:15-1= 14
Calculate the critical value
If the calculated t value is less than -2.145 or greater than 2.145, then reject the null hypothesis.
- x̅ = 123
- μ0 = 120
Calculated t statistic value less than the critical value, hence failed to reject null hypothesis ( H0). So, there is no significant difference between sample mean and population mean.