P-value vs. Critical Value Approach Both the p-value and the critical value methods are used to determine whether to reject or not a null hypothesis....
Both the p-value and the critical value methods are used to determine whether to reject or not a null hypothesis. However, there are key differences between the two approaches:
P-value:
The p-value is the probability of observing a test statistic as extreme as the one observed, assuming that the null hypothesis is true.
It tells you the probability of making a Type I error (rejecting the null when it's true) under the null hypothesis.
A low p-value (less than 0.05) suggests evidence against the null hypothesis, while a high p-value (greater than 0.05) suggests insufficient evidence to reject the null.
The p-value is a specific value for each observation, so it requires knowing the sample size and the specific test statistic value.
Critical value:
The critical value is the threshold used to determine the p-value.
It is based on the significance level (typically set to 0.05) and the degrees of freedom (which depends on the sample size).
A critical value tells you the maximum p-value you are willing to accept.
If the p-value is less than the critical value, you reject the null hypothesis.
If the p-value is greater than the critical value, you fail to reject the null hypothesis.
Example:
Imagine you're testing if the average height of women is equal to 6 feet. You randomly select 100 women and measure their heights. The average height is 6.1 feet, and the standard deviation is 0.5 feet.
Using the p-value approach, you calculate the probability of getting a test statistic as extreme as 6.1 feet with a sample size of 100, assuming the null hypothesis is true. This probability is very small (close to 0).
Based on the chosen significance level of 0.05, you reject the null hypothesis. This means that there is enough evidence to conclude that the average height of women is not equal to 6 feet.
Using the critical value approach, you set the critical value at 0.05 (the significance level).
Calculate the p-value for the observed test statistic. Since the p-value is smaller than 0.05, you fail to reject the null hypothesis. This means that there is not enough evidence to conclude that the average height of women is not equal to 6 feet