Describe the 8 steps in the process for hypothesis testing. Explain the decision criteria for rejecting the null hypothesis for both the p-value method and the critical value method.
Answer and Explanation:
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Step 1: State the null and alternative hypothesis. The hypothesis can be one tailed (right or left tailed) or two-sided.
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Step 2: Identify the sample evidence, that is, the sample mean, the population standard deviation and the sample size
Step 3: Choose the type of test to perform. If the sample size is large ( and the population standard deviation is known, a z-test is conducted. If the population standard deviation is unknown and its stated that the data is normally distributed, a z-test is conducted. Otherwise, a t-test is conducted when n<30 and the population standard deviation is unknown.
Step 4: Choose the level of significance to use in performing the hypothesis test and calculate the critical value. Commonly used is 5% level of significance.
Step 5: Compute the test statistics using the formula; or
Step 6: State the decision criteria. If the obtained test statistic sis greater than the critical value, the null hypothesis is rejected. This is the critical value method. For the p-value approach, if the p-value is less than the significance level, α, the null hypothesis is rejected. Otherwise, we fail to reject the null hypothesis.
Step 7: Make a decision by comparing the critical value and the test statistic if using the critical value method or comparing using the p-value and the significance level.
Step 8: Draw a conclusion based on the results of step 7. If the null hypothesis is rejected, it implies that there is sufficient evidence to support the claim being tested. If we fail to reject the null hypothesis, the data does not support the claim being tested.
The remaining problems refer to the following scenario:
A claim is made that the average salary for all jobs in Minnesota is less than $75,000. You are going to test the claim using and assume that your data is normally distributed and the population standard deviation is not known.
Write the null and alternative hypotheses symbolically and identify which hypothesis is the claim. Then identify if the test is left-tailed, right-tailed, or two-tailed and explain why.
Answer and Explanation:
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The type of hypothesis test is left tailed. The test is left tailed because the claim it to test if the average salary for all jobs in Minnesota is less than $75,000.
Identify and explain which test statistic you will use for your hypothesis test: z or t? Find the value of the test statistic.
Answer and Explanation:
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Since σ is unknown, and the sample standard deviation s is used in its place, then the test statistic follows a t-distribution with n-1 degrees of freedom. Since for this case, n is large ( then a t-distribution with n-1 degrees of freedom is almost identical to a standard normal distribution, so a Z-test statistic will be used instead of a t-test; ,
What is the critical value? Describe the rejection region.
Answer and Explanation:
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Since the alternative hypothesis has the less symbol, the hypothesis test is a left-tailed and we have one critical value. The critical value is , where is the z-value that cuts off an area α in the right tail of a standard normal density.
The significance level is α=0.05. Using the standard normal table, the critical value is thus . The null hypothesis is rejected if it less than the critical value. In this case, < and it lies in the rejection region, hence null hypothesis is rejected.
Using the critical value approach, should you reject the null hypothesis or not reject the null hypothesis? Explain. After making your decision, restate it in non-technical terms and make a conclusion about the original claim.
Answer and Explanation:
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The test statistic is which is in the rejection region since < . Thus, the null hypothesis that states that is rejected at α=0.05 and conclude that the average salary for all jobs in Minnesota is less than $75,000.
Calculate the p-value for this hypothesis test, and state the hypothesis conclusion based on the p-value. Does this match your results from the critical value method?
Answer and Explanation:
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The p-value obtained from the test is p=0.0054 which is less than the significance level, α=0.05. Therefore, the null hypothesis is rejected and conclude that there is evidence at 5% significance level that the average salary for all jobs in Minnesota is less than $75,000. The result obtained is similar to the one obtained when using critical method.
