t-Test (One Sample)

The t-Test (One Sample) tool is used to determine if the sample mean of a dataset is significantly different than a known or hypothesised population mean. It is commonly applied when the test statistic would follow a normal distribution if the value of a scaling term in the test statistic were known. When the scaling term is unknown and is replaced by an estimate based on the data, the test statistic (under certain conditions) follows a Student’s t distribution.

For example, it can be used to determine if there are significant differences between the mean rate of incidence of a disease and a hypothesised rate of incidence, or if the mean number of occupied dwellings in an area is significantly different to the known number.




For this worked example, we will compare the rates of cancer incidence per 100,000 people between 2006-2010 in Greater Brisbane with The Australian Institute of Health and Welfare’s (AIHW) estimated cancer incidence rate per 100,000 people for 2020.

Select the Greater Brisbane GCCSA as your area.

Select AIHW – Cancer Incidence and Mortality Across Regions (CIMAR) – Persons Incidence (SA3) 2006-2010 as your dataset, with the following variables:

  • SA3 Code
  • SA3 Name
  • All Cancers Combined – Age-standardised rate per 100,000



Once you have set up your data, open the t-Test (One Sample) tool (Tools → Statistical Analysis → t-Test (One Sample)). The input fields are as follows:

  • Dataset Input: The dataset containing the variables you would like to test. Select: AIHW – Cancer Incidence and Mortality Across Regions (CIMAR) – Persons Incidence (SA3) 2006-2010.
  • Variable: The variable we would like to test. Select: All Cancers Combined – Age-standardised rate per 100,000.
  • Population Mean: The known or hypothesised population mean that we are testing against. Select: 481.6.
  • Confidence Level: The confidence intervals for the calculation of means that will be undertaken as part of the test, ranging from 0.90 to 0.99. Select: 0.95.
  • Alternative Hypothesis: Specifies the alternative hypothesis. Select: two.sided.
    • two.sided: a priori assumption that the difference between the sample mean and the population mean is not equal to zero.
    • greater: a priori assumption that the sample mean is greater than the population mean.
    • less: a priori assumption that the sample mean is less than the population mean.

The input parameters are summarised in the image below, once complete click Run Tool.



Once the tool has run, click the Display button on the pop-up dialogue box that appears. This will open a window with the outputs of your t-Test (One Sample), which should look like the image below. The statistical significance is determined by the p-value. In this example, the p-value suggests that the differences between cancer incidence rates are significantly different since the p-value is lower than 0.05.

The output values are as follows:

  • t: The value of the t-statistic.
  • df: The degrees of freedom for the t-statistic.
  • p.value: The probability value for the t-statistic.
  • Confidence level: The confidence level that you selected for your test.
  • Confidence Interval: The confidence interval for the mean appropriate to the specified alternative hypothesis:
    • two.sided: The confidence intervals include lower and upper limits.
    • greater: The confidence intervals include lower limit and positive infinite.
    • less: The confidence intervals include negative infinite and upper limit.
  • mean of x: The mean of the input values.
  • null.value: The specified known or hypothesised population mean.
  • method: The type of t-test.

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