- Complete a one-sample [latex]z[/latex]-test for proportions from hypotheses to conclusions.
- Use a P-value to explain the conclusions of a completed [latex]z[/latex]-test for proportions.
Test Statistic
To draw a conclusion using data, first we have to calculate a test statistic. A test statistic is used in a hypothesis test to decide whether the data support or reject the null hypothesis.
test statistic ([latex]z[/latex]-statistic)
A test statistic measures the distance between the sample statistic and the null hypothesis value in terms of the hypothesized standard deviation of the null hypothesis value.
[latex]\text{test statistic}: z = \dfrac{\text{sample statistic }-\text{ null hypothesis value}}{\text{standard deviation of the null hypothesis value}}[/latex]
[latex]\text{test statistic}: z = \dfrac{\stackrel{ˆ}{p}-p_0}{\sqrt{\frac{p_0(1-p_0)}{n}}}[/latex]
where [latex]\stackrel{ˆ}{p}[/latex] is the sample statistics and [latex]p_0[/latex] is the null hypothesis value.
When the sample statistic is a proportion, the test statistic is also called a [latex]z[/latex]-statistic.
The [latex]z[/latex]-score (the test statistic) tells us how far the sample proportion is from the null hypothesis. The test statistic assesses how consistent the sample data collected are with the null hypothesis in a hypothesis test. We can use this statistic to find the probability of how likely it is that the data would have occurred by random chance.
Assuming the sample size is large enough ([latex]np \geq 10, n(1-p) \geq 10[/latex]), we can then use the normal distribution to calculate such probability, which we can use to make an inference to draw conclusions about the population parameter.