How to Find p Value from Test Statistic

P-values are widely used in statistics and are important for many hypothesis tests. But how do we find a p-value? The method can vary depending on the specific test, but there's a general process we can follow. In this article, we'll learn how to find the p-value, get an overview of the general steps for all hypothesis tests, and see a detailed example of how to calculate a p-value.

Hypothesis tests check if a claim about a population is true. This claim is called the null hypothesis (H0). The alternative hypothesis (Ha) is what we would believe if the null hypothesis is false. Knowing how to find the p-value is crucial in testing because it helps we decide if the null hypothesis is likely true or not.

Understanding p-value and Test Statistic

To understand more about p-value and test statistics read the article added below:

p-Value

The p-value is calculated using the test statistic's sampling distribution under the null hypothesis, the sample data, and the type of test being conducted (lower-tailed, upper-tailed, or two-sided test).

Test Statistic

A test statistic measures how closely our data matches the distribution predicted by the null hypothesis of the statistical test you’re using.

Steps to Find p-value from Test Statistic

Here are steps to help calculate the p-value for a data sample:

Step-1: State Null and Alternative Hypotheses

Start by looking at our data and forming a null and alternative hypothesis. For example, we might hypothesize that the mean "μ" is 10. Thus, the alternative hypothesis is that the mean "μ" is not 10. we can write these as:

H₀: μ = 10

H₁: μ ≠ 10

In these hypotheses:

• H0 is the null hypothesis.
• H1 is the alternative hypothesis.
• μ is the hypothesized mean.
• ≠ means does not equal.

Step-2: Use a t-test and its Formula

After setting our hypotheses, calculate the test statistic "t" using our data set. The formula is:

t = (x̄ - μ) / (s / √n)

where,

• t is the test statistic.
• x̄ is the sample mean.
• μ is the hypothesized mean.
• s is the standard deviation of the sample.
• n is the sample size.

Standard deviation measures how spread out the data points are in a set. It shows how close a data point is to the mean compared to other data points.

Step-3: Use a t-distribution table to find the p-value

After calculating "t," find the p-value using a t-distribution table, which we can find online. The table includes significance levels of 0.01, 0.05, and 0.1, which indicate how close the hypothesis is to the data set. To use the table, subtract 1 from our sample size "n."

For example:

n = 10

10 - 1 = 9

Use this number and our chosen significance level to find the corresponding value in the table.

If we have a one-tailed distribution, this value is our p-value. For a two-tailed distribution, which is more common, multiply this value by two to get our p-value.

Example Calculating p-value

Here’s an example of calculating the p-value based on a known set of data:

Emma wants to know if the average number of hours students study each week is 15 hours. She gathers data from a sample of students and finds that the sample mean is 13 hours, with a standard deviation of 3 hours. She decides to perform a two-tailed t-test to find the p-value at a 0.05 significance level to determine if 15 hours is the true mean. She forms the following hypotheses:

• H₀: μ = 15 hours
• H₁: μ ≠ 15 hours

After forming her hypotheses, she calculates the absolute value, or "|t|," of the test like this:

• t = (13 - 15) / (3 / √20)
• t = (-2) / (0.67082)
• t = -2.980
• |t| = 2.980

Using this t-value, she refers to a t-distribution table to locate values based on her significance level of 0.05 and her t-value of 2.980. She uses a sample size of 20 and subtracts 1 to get the degrees of freedom:

• 20 - 1 = 19

She then reviews her t-value of 2.980, which falls between the levels 0.005 and 0.001 on a t-distribution table. She averages 0.005 and 0.001 to get a value of 0.003. With a two-tailed test, she multiplies this value by 2 to get 0.006, which is the p-value for this test. Since the p-value is less than the 0.05 significance level, she rejects the null hypothesis and accepts the alternative hypothesis that the average number of hours students study each week is not 15 hours.

Using Statistical Software to Find p-value

P-Values can be calculated using p-value tables, spreadsheets, or statistical software like R or SPSS. we can find out how often the test statistic falls under the null hypothesis by using the test statistic and degrees of freedom (number of observations minus the number of independent variables).

The method to calculate a p-value depends on the statistical test you're using. Different statistical tests have different assumptions and produce different statistics. Choose the test method that best fits our data and the effect or relationship you're testing. The number of independent variables in our test affects the size of the test statistic needed to produce the same p-value.

No matter what statistical test we use, the p-value always indicates how often we can expect to get a test statistic as extreme or more extreme than the one from our test.

Practical Applications of p-value

P-Value is important in many engineering fields, from electrical engineering to civil engineering. It helps test prototype reliability, validate experiment results, and optimize systems, supporting statistically-informed decisions.

• Electrical Engineering: Electrical engineers use P-Values to test the efficiency of electrical devices, compare different models' performance, and validate results from complex circuit simulations.

• Civil Engineering: In civil engineering, P-Values help validate the strength of construction materials, assess new design methods' effectiveness, and analyze various structural designs' safety.