how to find p value from test statistic

One of the most mutual tests used in statistics is thet-test, which is frequently used to decide if a population mean is equal to some value.

For example, suppose nosotros want to know if the mean height of a certain species of constitute is equal to 15 inches. To exam this, we could collect a random sample of 20 plants, find the sample mean and sample standard deviation, and perform a t-test to make up one's mind if the mean height is actually equal to 15 inches.

The null and culling hypothesis for the exam are equally follows:

H₀: µ = 15

H_a: µ ≠ xv

The formula for the test statistic is:

t = (10-μ) / (s/√n)

where ten is the sample mean,μ is the hypothesized mean (in our instance information technology would be 15), due south is the sample standard departure, and n is the sample size.

In one case 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) and so we can refuse the null hypothesis and conclude that the mean height of the plants isnotequal to 15 inches.

Still, information technology's besides possible to approximate the p-value of the exam past hand using a t-Distribution table. In this postal service, nosotros'll explicate how to do so.

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

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

Solution:

Stride 1: State the null and culling hypotheses.

H₀: µ = 15

H_a: µ ≠ fifteen

Step 2: Observe the test statistic.

t = (ten-μ) / (s/√north) = (fourteen-xv) / (three/√20) = -ane.49

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

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

In the t-Distribution table beneath, nosotros need to look at the row that corresponds to "19" on the left-hand side and attempt to await for the accented value of our test statistic1.49.

Notice that ane.49 does non testify up in the table, simply it does fall betwixt the two values1.328and1.729.

Adjacent, we tin look at the two alpha levels at the top of the table that correspond to these two numbers. We run into that they are0.1and0.five.

This ways that the p-value for a ane-sided test is between 0.1 and 0.05. Let's telephone call it .075. Since our t-test is 2-sided, we need to multiply this value by 2. So, our estimated p-value is .075 * two = 0.15.

Pace 4: Draw a determination.

Since this p-value isnonless than our chosen alpha level of .05, we tin't turn down the nil hypothesis. Thus, we don't accept sufficient evidence to say that the true mean height of this species of plant is different than 15 inches.

Verifying the Results with a Figurer

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

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

Conclusion

We saw in this post that information technology's possible to estimate the p-value of a t-examination by hand using the t-Distribution table. However, in most scenarios you will never have to calculate the p-value past paw and instead you lot can utilise either statistical software like R and Excel, or an online computer to discover the exact p-value of the examination.

In most cases, especially in rigorous statistical studies and experiments, you volition want to use a calculator to find the exact p-value from a t-exam and then that you can be as accurate as possible, but it'due south skilful to know that you can still estimate the p-value from a t-test by hand if you lot absolutely need to.

Source: https://www.statology.org/how-to-calculate-a-p-value-from-a-t-test-by-hand/

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