Joel, I have not checked all your calculations, but the process you used looks good. I note that this is a problem where you are given a lot of raw data. Which technology did you use? I ask because problems where you have to manually enter a lot of raw data often lead to mistakes just in keying the data in.
Here is another MSL problem where you really have a lot to key in because there are 30 sample values each with 5 digits to enter.
I worked it in Excel and would like to point out two other things:
- One key is to recognize you are given the population standard deviation, sigma, so you should use the z-test.
- You must recognize the claim and state it both in words and math symbols. If you do that correctly, then doing the math become easier as does writing the conclusion.
Here, because we have a right-tail test, we must subtract the values we get from the tables or basic Excel functions from 1. Failing to recognize this results in the many, many mistakes I see in student quizzes with a P-value that is the wrong tail.
And if you know which hypothesis the claim is, writing the conclusion becomes a lot easier.
I would also note that in the image, I show how to get Excel to use the basic logic of hypothesis tests to “make” the Reject or Fail to Reject decision automatically using Excel functions AND and IF. On the calculators on my website, I take it one more step and get Excel to write the conclusion too. This is not difficult to do if you want to create your own Excel calculators which you can then use on quizzes and exams.
After you have completed the statistical analysis and decided to reject or fail to reject the Null hypothesis, you need to state your conclusion about the claim. To get the correct wording, you need to recall which hypothesis was the claim.
If the claim was the null, then your conclusion is about whether there was sufficient evidence to reject the claim. Remember, we can never prove the null to be true, but failing to reject it is the next best thing. So, it is not correct to say, “Accept the Null.”
If the claim is the alternative hypothesis, your conclusion can be whether there was sufficient evidence to support (prove) the alternative is true.
Use the following table to help you make a good conclusion.
The best way to state the conclusion is to include the significance level of the test and a bit about the claim itself.
For example, if the claim was the alternative that the mean score on a test was greater than 85, and your decision was to Reject then Null, then you could conclude:
“At the 5% significance level, there is sufficient evidence to support the claim that the mean score on the test was greater than 85.”
The reason you should include the significance level is that the decision, and thus the conclusion, could be different if the significance level was not 5%.
If you are curious why we say “Fail to Reject the Null” instead of “Accept the Null,” this short video might be of interest: Here
Here is a way to do a simple one-sample z test when we know the population standard deviation sigma. If this was a left tail test as indicated by an < in the alternative hypothesis, the standardized test statistic (z) of -1.753 falls in the rejection region below the z-critical of -1.645 and that would tell us to Reject the Null. The p-value for that same left-tail test is 0.0398 which is less than the significance level of 0.05, again telling us to reject the null.
Empirical Rule percentiles are the percentage of data below (to the left of) an x value. Use this Quick and Easy calculator to find percentiles when you are given the population mean and standard deviation and x values. In most intro stats classes, you will only be given x values whose z-scores are are integers. If a problem has x values whose z-scores are not integer values, you can find percentiles using the Normal tab on the workbook. If you would like a copy of this workbook, email me at drdawn@TheStatsFiles.com and let me know you have subscribed to my YouTube channel.
Question: The problem I struggled with this week is 8.1.27. It seemed like all the rest of the problems in the first chapter, but I got it wrong because I used “0” instead of the µ1-µ2 value. Not sure when I am supposed to use that value or how it is determined. I used the “view an example” feature to figure out I was supposed to put it in there for the homework, but I am worried about the quiz and not being able to recognize the situation it is needed.
Answer: When we are comparing two samples, whether means or proportions pay close attention to the claim. Here, the claim is that the difference in the two mean salaries is more than $5000. That requires the claim to be the alternative hypothesis since “more than” is a > operator, which is an inequality. So, Ha: μ1 – μ2 > $5000.
I know. You are saying that you don’t see a claim. But when a problem asks a question as this one does, that is the claim to be tested unless you find a more definitive claim later in the problem.
Note that you need to follow the “standard” format which is that we consider μ1– μ2 and not the reverse because that is the way the claim is stated in the problem: Region 1 is mentioned before Region 2. I think it might be less confusing if the problem had compared Alabama and Florida salaries, but the first entity (population) mentioned is logically μ1.
Here is the StatCrunch solution: [Read more…] about Pick-a-Problem: What difference does it make? 8.1.27