Rejection Region

Initial Post Instructions

After reviewing data from a sample, an inference can be made about the population. For example,

Find a data set on the internet. Some suggested search terms: Free Data Sets, Medical Data Sets, Education Data Sets.

  1. Introduce your Data Set and Cite the Source.
  2. What trends do you notice in your data set?
  3. Based on the trends and the history of your data set, make a claim. What kind of test (left, right, two tailed) would you have to complete?
  4. Explain the steps needed to complete the Hypothesis Test. What is needed?
Hypothesis Testing

Our desire with hypothesis testing is to determine if a claim regarding a population is correct. The answer to this question depends on many factors, some being 1) the sample size, and 2) the sample parameter, and 3) the variability of the samples.

Let’s look at an example. If an instructor believes s/he has a better way to teach statistics and measures success on final exam test scores we can use hypothesis testing to test if the "better way to teach statistics" really is "better." There must be some historical information from which the instructor wishes to improve. Let’s say that using the past 5 years of data, in which there really has been very little changes in the teaching methods, the final exam scores have averaged 75 with a standard deviation of 3 points. With the new methods, a sample of 15 students is taken and shows final exam scores are higher on the average, let’s say 80. Has the methods used by the instructor really shown improvement relative to the student’s ability to perform well on the final exam?

*** IMPORTANT ***

In the example above, we will set up two hypothesis: the null hypothesis and the alternate hypothesis. Now…….. (key point here)……. we always think the alternate hypothesis is correct!

 

We use Ho to represent the null hypothesis and Ha to represent the alternate hypothesis. Let “mu” equal the true mean of all statistics students on the final exam, which is actually the population mean. We think, based upon the sample of 15 students, that this population mean is now above 75. (Why? The sample mean is 80… which is above 75!)

So, Ho is “mu <= 75” and Ha is “mu > 75”. Remember that we always think the alternate hypothesis is correct.

Now, we need to figure out which one of these hypotheses is correct. We will assume the null hypothesis is correct and we’ll try to reach a conclusion that we should reject that assumption. That is, we will try to reject the null hypothesis. That means that we accept the alternate hypothesis, which is what we thought was correct at the very beginning!!!!

We will use a confidence level equal to 1-alpha to help us determine how far above 75 our sample mean needs to be in order to determine that the true population mean is above 75. Remember that we have a sample that suggests the mean is greater than 75, but the sample is not the actual population. The sample mean is 80 and we’re not sure that 5 point difference is definite. If we take another sample, that sample will undoubtedly have a different sample mean. If we took a third sample, it would undoubtedly have a sample mean different from the first two. So, based upon our one sample, how far about 75 must the sample mean be for us to decide that the true population mean is above 75? That’s the purpose of hypothesis testing.

 

Hypothesis testing provides us with a series of specific steps that lets us conclude which hypothesis is correct.

1st Step: Determine the null and alternate hypothesis. (We always think the alternate is correct.)

2nd Step: Determine the “rejection region” based upon alpha and the hypothesis from Step 1. If the alternate hypothesis is “not equal to” we have a two-tailed rejection region. If the alternate hypothesis is “less than” we have a left-tailed rejection region. If the alternate hypothesis is “greater than” we have a right-tailed rejection region.

3rd Step: Calculate the test statistic based upon the data obtained from the sample.

4th Step: Make a decision using test statistic from step 3 with conditions in step 2.

5th Step: Summarize decision in terms using “English” and not statistical speak!!

 

Math Notation

Class, using proper Math Notation is important. But, using correct notation in a “text” based environment, like discussion areas, can be a challenge.

The following notation is generally regarded as sufficient to convey mathematical symbols. Please use it as you wish.

To write “less than or equal to”, the correct notation is ≤. We can, however, use “<=” to convey this meaning using text.

To write “greater than or equal to”, the correct notation is ≥. We can use “>=” to convey this meaning using text.

To write “not equal to”, the correct notation is ≠. We can use “not=” to convey this meaning using text.

Solution:

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