Question

Use the definition of the \(P\) -value to explain the following: a. Why \(H_{0}\) would be rejected if \(P\) -value \(=0.0003\) b. Why \(H_{0}\) would not be rejected if \(P\) -value \(=0.350\)

Short answer

A low \(P\)-value (\(0.0003\)) is strong evidence to reject the null hypothesis (\(H_{0}\)), while a high \(P\)-value (\(0.350\)) does not allow to reject the \(H_{0}\) hypothesis since there's a higher probability to observe such an extreme result considering the null hypothesis is true.

Step-by-step solution

Understanding the Null Hypothesis

The null hypothesis (\(H_{0}\)) is a general statement or default position that there is no relationship between two measured phenomena, or no association among groups. Rejecting or not rejecting the null hypothesis depends on the \(P\)-value and the chosen level of significance.

Explain Why \(H_{0}\) is Rejected if \(P\)-value \(=0.0003\)

If the \(P\)-value is \(0.0003\), it means that there's only a \(0.03\%\) chance of seeing a result as extreme as the observed one, assuming the null hypothesis is true. Since this percentage is less than typically agreed upon significance level \(5\%\) (\(0.05\)), the null hypothesis \(H_{0}\) is rejected. The smaller the \(P\)-value, the stronger the evidence against the null hypothesis.

Explain Why \(H_{0}\) is Not Rejected if \(P\)-value \(=0.350\)

If the \(P\)-value is \(0.350\), it means that there's a \(35\%\) chance of seeing a result as extreme as the observed one, assuming the null hypothesis is true. Since this probability \(0.350\) is higher than the commonly used significance level, which is \(5\%\) or \(0.05\), there is not enough evidence to reject the null hypothesis \(H_{0}\).