# How to Find P-Value

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• 0:00 Tests of Significance
• 0:56 P-Value Defined
• 1:45 Finding the P-Value
• 3:07 The Critical Value
• 3:58 What Significance Tells Us
• 5:05 Lesson Summary

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Lesson Transcript
Instructor: Tracy Payne, Ph.D.

Tracy earned her doctorate from Vanderbilt University and has taught mathematics from preschool through graduate level statistics.

When is a number so much bigger or smaller than another that it should raise some eyebrows? Tests of significance help make such determinations. This lesson explains the p-value in significance tests, how to calculate them, and how to evaluate the results.

## Tests of Significance

Imagine that you want to be the new point guard of your basketball team, but before you try out for the position, you want to make sure you have, pun intended, a real shot at achieving your goal. You shoot 20 free throws and make 12 of them; that's a 60% accuracy rate. You want to know if your accuracy rate, or the observation, is about the same or different than the team's accuracy rate, or the population statistic; enough to replace the old point guard.

You can do a test of significance to ascertain if your accuracy rate is significantly different from that of the team. A significance test measures whether some observed value is similar to the population statistic, or if the difference between them is large enough that it isn't likely to be by coincidence. When the difference between what is observed and what is expected surpasses some critical value, we say there is statistical significance.

## P-Value Defined

A standard normal distribution curve represents all of the observations of a single random variable such that the highest point under the curve is where you would expect to find values closest to the mean and values least likely to be observed in the smallest part under the curve.

The p-value is the probability of finding an observed value or a data point relative to all other possible results for the same variable. If the observed value is a value most likely to be found among all possible results, then there is not a statistically significant difference. If, on the other hand, the observed value is a value among unlikely values to be found, then there is a statistically significant difference. The smaller the probability associated with the observed value, the more likely the result is to be significant.

## Finding The P-Value

To find the p-value, or the probability associated with a specific observation, you must first calculate the z score, also known as the test statistic.

The formula for finding the test statistic depends on whether the data includes means or proportions. The formulas we'll discuss assume a:

1. Single sample significance test
2. Normal distribution
3. Large sample size.

When dealing with means, the z score is a function of the observed value (x-bar), population mean (mu), standard deviation (s), and the number of the observations (n).

When dealing with proportions, the z score is a function of the observed value (p-hat), proportion observed in the population (p), probability of successful outcome (p), probability of failure (q = 1 - p), and the number of trials (n).

After calculating the z score, you must look up the probability associated with that score on a Standard Normal Probabilities Table. This probability is the p-value or the probability of finding the observed value compared to all possible results. The p-value is then compared to the critical value to determine statistical significance.

## The Critical Value

The critical value, or significance level, is established as part of the study design and is denoted by the Greek letter alpha. If we choose an alpha = 0.05, we are requiring an observed data point be so different from what is expected that it would not be observed more than 5% of the time. An alpha equaling 0.01 would be even more strict. In this case, a statistically significant test statistic beyond this critical value has less than a 1 in 100 probability of occurring by chance.

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