The logic and hypotheses of NHST

Introduction to Hypothesis Testing with One Sample

One job of a statistician is to make statistical inferences about populations based on samples taken from the population. Confidence intervals are one way to estimate a population parameter. Another way to make a statistical inference is to make a decision about a parameter. For instance, a car dealer advertises that its new small truck gets 35 miles per gallon, on average. A tutoring service claims that its method of tutoring helps 90% of its students get an A or a B. A company says that women managers in their company earn an average of $60,000 per year.

A statistician will make a decision about these claims. This process is called “hypothesis testing.” A hypothesis test involves collecting data from a sample and evaluating the data. Then, the statistician makes a decision as to whether or not there is sufficient evidence, based upon analyses of the data, to reject the null hypothesis.

In this chapter, you will learn about the steps of null hypothesis significance testing (NHST) and the errors associated with NHST.

Hypothesis testing consists of two contradictory hypotheses or statements, a decision based on the data, and a conclusion. To perform a hypothesis test, a statistician will:

1. Set up two contradictory hypotheses that are mutually exclusive and exhaustive.

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2. State a decision criterion for determining what data would lead them to reject the null hypothesis or not.

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3. Collect sample data (in homework problems, the data or summary statistics will be given to you).

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4. Determine the correct distribution to perform the hypothesis test.

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5. Analyze sample data by performing the calculations that ultimately will allow you to reject or decline to reject the null hypothesis.

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6. Make a decision and write a meaningful conclusion.

Based on sample evidence, a procedure for determining whether the hypothesis stated is a reasonable statement and should not be rejected, or is unreasonable and should be rejected. This starts with specifying the null and alternative hypothesis, which we will go into more detail on below.

Null and Alternative Hypotheses

The actual test begins by considering two hypotheses. They are called the null hypothesis and the alternative hypothesis. These hypotheses contain opposing viewpoints that are mutually exclusive and exhaustive.

H₀: The null hypothesis: It is a statement about the population that either is believed to be true or is used to put forth an argument unless it can be shown to be incorrect beyond a reasonable doubt. This is usually, but not always, a statement of (null) no effect.

H_a: The alternative hypothesis: It is a claim about the population that is contradictory to H₀ and what we conclude when we reject H₀. This is usually, but not always, a statement that there is an effect and is based on the theory being tested.

Since the null and alternative hypotheses are contradictory, you must examine evidence to decide if you have enough evidence to reject the null hypothesis or not. As the null and alternative hypotheses are mutually exclusive and exhaustive, if we reject the null hypothesis, then we consider this support for the alternative hypothesis. The evidence needed to examine if we reject the null hypothesis is in the form of sample data.

After you have determined which hypothesis the sample supports, you make adecision. There are two options for a decision. They are “reject H₀” if the sample information favors the alternative hypothesis or “do not reject H₀” or “decline to reject H₀” if the sample information is insufficient to reject the null hypothesis.

Mathematical Symbols Used in H₀ and H_a:

Note

H₀ always has a symbol with an equal in it. H_a never has a symbol with an equal in it. The choice of symbol depends on the wording of the hypothesis test. However, be aware that many researchers (including one of the co-authors in research work) use = in the null hypothesis, even with > or < as the symbol in the alternative hypothesis. This practice is acceptable because we only make the decision to reject or not reject the null hypothesis.

This video provides an additional explanation of simple hypothesis testing.

Concept Review

In a hypothesis test, sample data is evaluated in order to arrive at a decision about some type of claim. If certain conditions about the sample are satisfied, then the claim can be evaluated for a population. In a hypothesis test, we: Evaluate the null hypothesis, typically denoted with H₀. The null is not rejected unless the hypothesis test shows otherwise. The null statement must always contain some form of equality (=, ≤ or ≥). Always write the alternative hypothesis, typically denoted with H_a or H₁, using less than, greater than, or not equals symbols,( i.e., ≠, >, or <). If we reject the null hypothesis, then we can assume there is enough evidence to support the alternative hypothesis. Never state that a claim is proven true or false. Keep in mind the underlying fact that hypothesis testing is based on probability laws; therefore, we can talk only in terms of non-absolute certainties.

Formula Review

H₀ and H_a are contradictory hypotheses that are mutually exclusive and exhaustive.

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