How do you find p-value from test statistic?

One of the most common tests used in statistics is thet-test, which is oftenused to determine if a population mean is equal to some value.For example, suppose we want to know if th

One of the most common tests used in statistics is thet-test, which is oftenused to determine if a population mean is equal to some value.

For example, suppose we want to know if the mean height of a certain species of plant is equal to 15 inches. To test this, we could collect a random sample of 20 plants, find the sample mean and sample standard deviation, and perform a t-test to determine if the mean height is actually equal to 15 inches.

The null and alternative hypothesis for the test are as follows:

H0: µ = 15

Ha: µ  15

The formula for the test statistic is:

t = (x-μ) / (s/n)

wherexis the sample mean,μ is the hypothesized mean (in our example it would be 15), s is the sample standard deviation, and n is the sample size.

Once we know the value oft, we can use statistical software or an online calculator to find the corresponding p-value. If the p-value is less than some alpha level (common choices are .01, .05, and .10) then we can reject the null hypothesis and conclude that the mean height of the plants isnotequal to 15 inches.

However, its also possible to estimate the p-value of the test by hand using a t-Distribution table. In this post, well explain how to do so.

Example: Calculating the p-value from a t-test by hand

Problem: Bob wants to know if the mean height of a certain species of plant is equal to 15 inches. To test this, he collects a random sample of 20 plants and finds that the sample mean is 14 inches and the sample standard deviation is 3 inches. Conduct a t-test using a .05 alpha level to determine if the true mean height for the population is actually 15 inches.

Solution:

Step 1: State the null and alternative hypotheses.

H0: µ = 15

Ha: µ  15

Step 2: Find the test statistic.

t = (x-μ) / (s/n) = (14-15) / (3/20) = -1.49

Step 3: Find the p-value for the test statistic.

To find the p-value by hand, we need to use the t-Distribution table with n-1 degrees of freedom. In our example, our sample size is n = 20, so n-1 = 19.

In the t-Distribution table below, we need to look at the row that corresponds to 19 on the left-hand side and attempt to look for the absolute value of our test statistic1.49.

Notice that 1.49 does not show up in the table, but it does fall between the two values1.328and1.729.

Next, we can look at the two alpha levels at the top of the table that correspond to these two numbers. We see that they are0.1and0.5.

This means that the p-value for a one-sided test is between 0.1 and 0.05. Lets call it .075. Since our t-test is two-sided, we need to multiply this value by 2. So, our estimated p-value is .075 * 2 = 0.15.

Step 4: Draw a conclusion.

Since this p-value isnotless than our chosen alpha level of .05, we cant reject the null hypothesis. Thus, we dont have sufficient evidence to say that the true mean height of this species of plant is different than 15 inches.

Verifying the Results with a Calculator

We can plug our test statistictand our degrees of freedom into an online p-value calculator to see how close our estimated p-value was to the true p-value:

The true p-value is0.15264, which is pretty close to our estimated p-value of0.15.

Conclusion

We saw in this post that its possible to estimate the p-value of a t-test by hand using the t-Distribution table. However, in most scenarios you will never have to calculate the p-value by hand and instead you can use either statistical software like R and Excel, or an online calculator to find the exact p-value of the test.

In most cases, especially in rigorous statistical studies and experiments, you will want to use a calculator to find the exact p-value from a t-test so that you can be as accurate as possible, but its good to know that you can still estimate the p-value from a t-test by hand if you absolutely need to.