How to do a hypothesis test correctly everytime!

How to Do Hypothesis Tests
The Easy Way!
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It is critical that you identify what is being claimed. If you tend to skim-read, I suggest you actually read the problem out loud so that you have to pronouce every word. Granted, this may get you some looks if you are in a coffee shop or library quiet space. But I find it extremely helpful.

If you do not find the claim, it is very likely you will not get the problem correct. 

Example: 

The local school board has been concerned about the effectiveness of policies of the superintendent who was hired two years ago. On a state mandated standardized math test, the average score of 10th grade students in the state was 260 points with a standard deviation of 15 points. The superintendent believes 10th grade students in her district will score higher than the state average. The Board asked the Superintendent to provide some evidence the local students will do well on the test. 50 randomly selected 10th grade students in the district took the exam early. Does the data from the 50 students who took the early exams support the superintendent’s belief about how the local students will perform on the exam?

Sometimes the problem will actually use the word “claim” and that definitely helps. But more often, students must find phrases such as “believes that,” “wants to test whether,” “can you conclude that…” .

In the example above, you should get that the claim is:

“The superintendent believes the 10th grade students in her district will score higher than the state average.”

“The superintendent believes the 10th grade students in her district will score higher than the state average.”

The point of this step is to make sure you identify words that identify the population parameter of interest and the math operator in the claim.

In the example, the claim is about the average score on the state test. Average is the same as the mean, so the parameter of interest is the population mean “mu” which has the small Greek letter µ as its math symbol.

The math operator is signaled by the phrase “score higher” which is the same as “more than” or “greater than” and is the symbol >.

Since the state average on the test was 260 points, we get µ > 260 for the claim stated as a math equation.

Unlike more advanced stats courses designed to prepare future researchers to conduct effective research, our intro stats course, like many others, does not suggest or recommend that the research hypothesis, the claim, always be the alternative. Recall that the reasons some authors give that guidance is that it is not possible to “prove” your research hypothesis if it is the Null. In a correctly run null hypothesis significance test, we can never accept or prove the Null to be true. We can only fail to reject it if the evidence for the Alternative is not sufficient to support the Alternative hypothesis. So, if you want to prove your research hypothesis is correct, it must be the Alternative. See here for more info on this topic. 

IMPORTANT: The claim can be either the Null or the Alternative hypothesis. The math operator in the claim determines which type of hypothesis it is. 

Using the rules of the Null Hypothesis Significance Test (NHST) we follow in our course, as do most intro stats courses,

  • The Null and Alternative hypotheses account for all possiblities in the sample space negative infinity to positive infinity.
  • The Null hypothesis is always the “no difference,” “no change,” nothing happening here” alternative. That means it is always a form of equality. It does not have to be a pure equality, which is the equal sign, =. The math operator in the Null must be either <=, =, or >=.
  • The math operator in the Alternative must always be a form of inequality: <, not =, or >.
  • The Null and Alternative are complements. If one is true, the other must be false.
    • If the Null math operator is =, the Alternative must be not = or ≠.
    • If the Null math operator is <= or ≤, the Alternative must be >.
    • If the Null math operator is >= or ≥, the Alternative must be <.

The claim in the example is µ > 260. Because the math operator is a form of inequality, the claim must be the Alternative hypothesis. We use the convention that the Alternative is identified by H1 or Ha. Thus, our claim is Haμ > 260.

The Null must be the complement of the Alternative. The Null is identified as H0. Thus the Null is H0μ  260. This is logical since if the Alternative is true, the Null must be false because between them, they account for all possibilities in the sample space. The Alternative includes all values greater than 260 while the Null includes all values equal to or smaller than 260.

The conventional way to write this is: 
H0μ  260 
Haμ > 260 (Claim)

 IMPORTANT: The claim can be either the Null or the Alternative hypothesis. The math operator in the claim determines which type of hypothesis it is. 

Using the rules of the Null Hypothesis Significance Test (NHST) we follow in our course, as do most intro stats courses,

  • The Null and Alternative hypotheses account for all possibilities in the sample space, negative infinity to positive infinity.
  • The Null hypothesis is always the “no difference,” “no change,” nothing happening here” alternative. That means it is always a form of equality. It does not have to be a pure equality, which is the equal sign, =. The math operator in the Null must be LaTeX: \leLaTeX: ==, or LaTeX: \ge.
  • The math operator in the Alternative must always be a form of inequality: <, not =, or >.
  • The Null and Alternative are complements. If one is true, the other must be false.
    • If the Null math operator is =, the Alternative must be not = or ≠.
    • If the Null math operator is <= or ≤, the Alternative must be >.
    • If the Null math operator is >= or ≥, the Alternative must be <.

The claim in the example is µ > 260. Because the math operator is a form of inequality, the claim must be the Alternative hypothesis. We use the convention that the Alternative is identified by H1 or Ha. Thus, our claim is Ha: μ > 260.

The Null must be the complement of the Alternative. The Null is identified as H0. Thus the Null is H0: μ ≤ 260. This is logical since if the Alternative is true, the Null must be false because, between them, they account for all possibilities in the sample space. The Alternative includes all values greater than 260 while the Null includes all values equal to or smaller than 260.

The conventional way to write this is: 
H0: μ ≤ 260 
Ha: μ
 > 260 (Claim)

Words to Symbols Table

The table is similar to others you may find in textbooks or online, but I have added all the keywords/phases I have found in word problems in teaching my intro stats classes for the last seven years.

Read the problem carefully, looking for key phrases and words which will identify the claim. Look for those key phrases in the table below to determine if the claim is the null or alternative hypotheses. It can be either. Because the null and alternative must be complements, finding one will also give you the other as they will be in the same row.

For an example, one problem includes this sentence: “The test results prompts a state school administrator to declare that the mean score for the state’s 8th graders is more than 260.” The “more than” phrase is a key to recognizing the claim is “more than” which is in the Alternative Hypothesis column in the Right-tail Row 3. The Math Operator in that cell is >. Thus, the claim is: µ > 260. The math operator in the claim always indicates the tail of the test, e.g. a > symbol points to the right, so this is a right tail test.

In the center column on the Right-tail row, you see the two hypotheses statements written in mathematical format. You just need to substitute the claimed value for x to complete them: H0: μ 260, Ha: μ > 260.

In this  (and most) intro stats course, we have limited the types of hypothesis tests we run. They are classified by the number of samples and the population parameter involved. I find most students do better if they look at the number of samples first.

Note: Links to individual tests coming soon.

One-sample/One variable/One Population tests: Means or Proportions

  • Tests for the population mean:
    • one-sample z-test for population mean
    • one-sample t-test for population mean
  • Tests for the population proportion:
    • Two possible outcomes – one-sample z-test for population proportion
    • More than two possible outcomes – Chi-square Goodness of Fit test

Two-sample/One variable tests: Means or Proportions

  • Tests for the difference between population means.
    • Two-sample z-test for population means
    • Two-sample t-test for population means
  • Tests for the difference between population proportions.
    • Two possible outcomes: Two-sample z-test for population proportions

Multiple-Sample/multiple variable/multiple population tests

  • Test for relationships between two or more quantitative variables.
    • Regression (One Independent variable and One Dependent variable)
    • One-way ANOVA: One variable and two or more populations (not covered in BUS 233)
    • Two-way ANOVA: Two independent variables and one dependent variable (not covered in BUS 233)
  • Test for relationships between categorical variables
    • Chi-square Test for Independence: Two variables – One population.
    • Chi-square Text for Homogeneity: One variable – Two or more populations. (not covered in BUS 233)

Dawn E. Wright, Ph.D.

Dawn E. Wright, Ph.D.

After you have completed the statistical analysis and decided to reject or fail to reject the Null hypothesis, you need to state your conclusion. Remember, the point of the hypothesis test was to  determine if the evidence collected supported the claim. We tested the evidence against the null hypothesis to see if it was different enough that we could decide to reject the null. But our conclusion is about the claim. To state a correct conclusion, you need to recall which hypothesis was the claim.

If the claim is 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. 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

Dawn E. Wright, Ph.

The tail of the hypothesis test is determined by looking at the math operator in the alternative hypothesis. I like to think of the direction in which the math operator is pointing. A less than operator, <, seems to be pointing to the left in my mind. Thus, it indicates a left tail test. A greater than symbol, >, points right and indicates a right tail test. The not equal, ≠, does not point either way, so that clues me to think of a two-tail test.

Use this table to help you remember this:

For a significance level, alpha, of 0.05, we put all of alpha in a left or right tail test. Thus, the probability we use to find the critical value is 0.05. We find the critical values of z to be -1.645 for the left tail test and +1.645 for the right tail test. But in the two-tail test, we put half of alpha, or 0.025 in this example, in each tail. Using those probabilities, we find the critical values of -1.96 and +1.96.

https://licensebuttons.net/l/by-sa/3.0/88x31.png Dawn E. Wright, Ph.D.