I have been making Excel-based “calculators” to help some of my students who are finding other technology limiting or difficult to use. Currently, I have seven up on this site under the BUS 233 tab. Check them out here. This is the Two-sample z-test for the difference between proportions.
Read the problem looking for keywords and values:
- What type of variable is the focus of the problem? Is it quantitative, e.g. a mean, or categorical, e.g. a proportion or percent?
- How many variables are of concern?
- Is(are) the population(s) standard deviation, sigma, given?
- Are sample variances equal or are you instructed to assume they are equal?
- What is(are) the sample size(s)?
- Is a claim or proposition mentioned?
- Are you given summary statistics or raw data?
At many golf clubs, a teaching professional provides a free 10-minute lesson to new customers. A golf magazine reports that golf facilities that provide these free lessons gain, on average, $2100 in green fees, lessons, or equipment expenditures. A teaching professional believes that the average gain exceeds 2,100, and collects data (shown below in the solution examples) on the gain from 15 clubs in his area. At the 0.05 level of significance, does the data support the claim?
What should you glean from reading the problem statement? [Read more…] about Recognizing problem types: Hypothesis tests
I get this question a lot in my BUS 233 Business Statistics course. It comes when students see a problem similar to the following in their homework or on a quiz:
In a sample of 1000 home buyers, you find that 457 home buyers found their real estate agent through a friend. At α=0.08, can you reject the claim that 50% of home buyers find their real estate agent through a friend?
a) Write the claim mathematically and identify H0 and Ha. (b) Find the critical value(s) and identify the rejection region(s). (c) Find the standardized test statistic. (d) Decide whether to reject or fail to reject the null hypothesis.
If you run into a problem which asks you to find the standardized test statistic but does not give you the standard deviation, it is probably a proportion problem and this one is just that.
Ho: p = 0.50; Ha: p ≠ 0.50. [Read more…] about No Standard Deviation? How do I get the standardized test statistic?
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:
The Null and Alternative
The most common problem I noticed on this assignment was caused by failing to properly identify the appropriate null and alternative hypotheses. In part, this is due to the Evans text’s somewhat confusing explanation of how to do this – the “burden of proof” approach. There is a much simpler approach that always works.
First, make sure you closely read the problem statement looking for key words and phrases. This table may help:
The null hypothesis always [Read more…] about Setting up Hypothesis Tests