One-Sample Hypothesis Test for Proportions: Learn It 4

  • 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.

Making a Decision Based on P-value and Significance Level

A P-value can assist us in determining whether or not we have evidence to reject the null hypothesis. Once a P-value is calculated, we compare it to the significance level in order to decide whether we have enough evidence to reject the null hypothesis in favor of the alternative hypothesis.

In statistics, we establish a “cut-off” value. There is no absolute cut-off value, but typically we use [latex]5\%[/latex].

This [latex]5\%[/latex] represents the extreme areas under the curve, which means they represent unusual values. We compare the P-value to [latex]\alpha[/latex], which is the significance level of the test.

significance level

The significance level, [latex]\alpha[/latex], is the cut-off for P-values at which we have enough evidence to reject the null hypothesis.

Typically, small significance levels such as [latex]1\%[/latex], [latex]5\%[/latex], or [latex]10\%[/latex] are used in hypothesis testing.

 

In order to make a claim about the null hypothesis, we write [latex]\alpha[/latex] as a decimal and compare it to the P-value, as follows:

  • If P-value [latex]\leq \alpha[/latex], we have enough evidence to reject the null hypothesis, and we have convincing evidence to support the alternative hypothesis.
  • Otherwise, we fail to reject the null hypothesis or do not reject the null hypothesis, and we do NOT have convincing evidence to support the alternative hypothesis.
    • When we fail to reject a null hypothesis, it does not mean there is support in favor of the null hypothesis. Instead, this means that we just did not see enough evidence to be convinced that the null hypothesis is not true.
Once we decide whether we have enough evidence to reject a null hypothesis, we write a statement in the context of the original question asked in order to describe the outcome of the hypothesis test. It is important to remember that we never prove that a null hypothesis is true; we only conclude that the sample data collected either do or do not support the alternative hypothesis.

If we are rejecting the null hypothesis, we can write, “Since our P-value is less than __%, there is enough evidence to suggest that [rephrase the alternative hypothesis]”. If we are failing to reject the null hypothesis, we can write, “Since our P-value is more than __%, there is insufficient evidence to suggest that [rephrase the alternative hypothesis].”

A note on P-values: Though P-values are widely used, there are some limitations on their use and significant debate as to their reliability. Recall that a P-value is calculated based on one sample of data collected, and, as a result, it may be difficult to obtain a similar P-value upon replication of an experiment. Additionally, P-values can be manipulated by increased sample size. In some studies, instead of using a P-value to answer the binary question (“Is there or is there not statistical significance?”), it may be better to consider the effect size, which answers the question “How strong is the effect in the sample?” For example, in reporting the effect size, a researcher explains by how much a treatment works rather than just if it works. Additionally, in using effect sizes, quantitative comparisons between the results of different studies can be made.