Critical Value Method

Now, let’s learn how to find the critical values for the critical region in the distribution and make the final decision of rejecting or failing to reject the null hypothesis.

(Note: In the video below, the graph showing the distribution of average sales data at 1:06 incorrectly displays 370.6 as the sample mean instead of 370.16. Also, it would be σ¯x=15 instead of σ=15at 3:41)

Before you proceed with finding the Zc and finally the critical values, let’s revise the steps performed in this method till now.

  • First, you define a new quantity called α, which is also known as the significance level for the test. It refers to the proportion of the sample mean lying in the critical region. For this test, α is taken as 0.05 (or 5%).
  • Then, you calculate the cumulative probability of UCV from the value of α, which is further used to find the z-critical value (Zc) for UCV.

Attempt the following questions before you go ahead and learn the remaining steps in this method.

After formulating the hypothesis, the steps you have to follow to make a decision using the critical value method are as follows.

  • Calculate the value of Zċ from the given value of α (significance level). Take it a 5% if not specified in the problem.
  • Calculate the critical values (UCV and LCV) from the value of Zċ.
  • Make the decision on the basis of the value of the sample mean x with respect to the critical values (UCV AND LCV).

You can download the z-table from the attachment below. It will be useful in the subsequent questions.

Let’s solve the following problem stepwise to consolidate your learning on how to make a decision about any hypothesis.

A manufacturer claims that the average life of its product is 36 months. An auditor selects a sample of 49 units of the product, and calculates the average life to be 34.5 months. The population standard deviation is 4 months. Test the manufacturer’s claim at 3% significance level using the critical value method.

First, you need to formulate the hypotheses for this two-tailed test, which would be:

                                   H₀:μ = 36 months and H₁: μ ≠ 36 months

Now, you need to follow the three steps to find the critical values and make a decision.

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