We are starting true tests of hypotheses and it is critical to get off on the right foot. I will add some useful info here shortly, but first I am sharing a question on a homework problem that is a good example of what you will see in the second half of the course.
What key information should you glean from reading the problem?
- This is a test for a single factor, the population mean, μ, which is a quantitative and not a categorical/qualitative variable.
- We are given the Claim that the mean, μ, equals (=) 50. Because the null hypothesis, Ho, always contains a form of equality (≤, =, ≥), the claim is the null hypothesis which is known as the ‘No difference’ hypothesis.
- The alternative hypothesis, Ha, is always the complement of the null. This means it always is a form of inequality, (<, ≠, >).
- The math operator in the alternative hypothesis always determines the ‘tail’ of the test we run. In this case, because the alternative contains the not = (≠) math operator, we have a two-tailed test.
- Because we have a two-tailed test, we must put half of alpha, α, in each tail to find the critical values. In this case, α =.06, α/2 = .03. You use either the tables or technology to find the critical value. I recommend you choose StatCrunch.
- Because we know sigma, the population standard deviation, and we are interested in a single quantitative variable, the mean, this is a z-test for the mean.
Now, how do we solve this problem?
- Using the StatCrunch normal (z) distribution calculator, and because we have a two-tailed test, we put the value of c = 1 – α = 0.94 in the area field of the “Between” option in the calculator to find the two critical values of z to be +/- 1.88. Note: if the area in the middle is 0.94, then there is 0.03 (α/2) in each tail.
- Using the Stat>Z-Stats>One Sample>with Summary (since we do not have raw data) sequence, we get the “One Sample Z Summary” dialog box.
- Put the sample mean, x-bar, in the “Sample Mean” box.
- Put the population sigma in the “Standard Deviation” box.
- Put the sample size n in the “Sample Size” box.
- Make sure the Hypothesis test for μ is selected in the “Perform:” area.
- Put the null hypothesis value of 50 in the “Ho: μ=” drop down box.
- Select the ≠ symbol in the “Ha: μ” box.
- Click “Compute!”
- The answer box (with the Options button) restates the test we have run. The standardized test statistic, Z-Stat, is -1.01, rounded to two decimal places.
- Because the standardized test statistic, -1.01, does not fall within either rejection area, we Fail to Reject the Null hypothesis and conclude “At the 6% significance level, there is not enough evidence to reject the claim.”
- Also, because the p-value of 0.314 is greater than the significance level of 0.06, we come to the same conclusion – Fail to Reject the Null, which is also the claim in this problem.
Here is a composite screenshot of the StatCrunch solution: