Session 16 Notes, Section 6-3

Hypothesis Testing:  Z Tests for Means (P-value Method)

In the previous session, you learned the "Classical Method" using the Z test for means.  The contemporary method for testing for means is the P-value method.  In this session you will learn the P-value method.  As you learn this method, keep in the classical method steps as your learn the steps for the P-value method.

At the end of this session you will be able to:

• Understand how the P-value is used
  • example
• State the five steps used in the classical method of hypothesis testing.
• State the five steps used in the P-value method of  hypothesis testing.

If you have learned all of these objectives, then close this window to return to where you were.

P-value Method for Large Samples

• The P-value (or probability value) is the probability of getting a sample statistic (such as the mean) or a more extreme sample statistic in the direction of the alternative hypothesis when the null hypothesis is true.
• The P-value is the actual area under the standard normal distribution curve (or other curve depending on what statistical test is being used) representing the probability of a particular sample statistic or a more extreme sample statistic occurring if the null hypothesis is true.  The P-value is  the associated "rejection area" associated with a particular sample statistic.

Decision Rule When Using a P-Value

• If there is a P-value ≤ α
  • reject the null hypothesis
• If there is a P-value > α
  • fail to reject the null hypothesis

Steps for solving Hypothesis-Testing Problems using the P-Value Method

YOU NEED TO KNOW THE STEPS!

• Step 1 State the hypothesis, and identify the claim.
• Step 2 Compute the test value using z  = (X - μ) /( σ/√n).
• Step 3 Find the P-value.
• Step 4 Make the decision to reject or fail to reject the null hypothesis.
• Step 5 Summarize the results.

Return to objectives

Example.

A researcher wishes to test the claim that the average age of lifeguards in a coastal town is greater than 25 years. She selects a sample of 32 guards and finds the mean of the sample to be 25.2 years, with a standard deviation of 2 years. Find the P-value. Is there evidence to support the claim at = 0.05?

• Review the steps for solving Hypothesis-testing problems using the P-value method. Follow them step by step.
• Step 1: State the hypotheses and identify the claim.
  • H₀:  µ ≤ 25  and  H₁: µ > 25 (claim)
• Step 2: Compute the test value using z  = (X - μ) /( σ/√n).
• Step 3: Find the P-value.
  • The corresponding area under the normal distribution for z = 0.57 is 0.2157. Subtract this value for the area from 0.5000 to find the area in the right tail.
  • 0.5000 - 0.2157 = 0.2843
  • Hence, the P-value is 0.2843
• Step 4: Make the decision.
  • Since the P-value is not less than 0.05, the decision is not to reject the null hypothesis.
• Step 5: Summarize the results.
  • There is not enough evidence to support the claim that the average age of lifeguards in that coastal town is greater than 25 years.

  Return to objectives

  ---

State the five steps used in hypothesis testing (Traditional Method).

• Step 1 State the hypothesis, and identify the claim.
• Step 2 Find the critical value from the appropriate table.
• Step 3 Compute the test value using z  = (X - μ) /( σ/√n).
• Step 4 Make the decision to reject or not reject the null hypothesis.
• Step 5 Summarize the results.

Return to objectives

When you finish these notes, then close this window to return to where you were.