Lesson 9.1 Hypothesis Tests in General

Hypothesis Testing

Hypothesis tests, like CIs, are a part of inferential statistics, the science of drawing statistical conclusions from specific data. A hypothesis test, called a test of significance, attempts to answer the question, "Could the data have occurred purely by chance?"

Hypothesis testing is done constantly in medicine, business, education, polling, government, and the hard and soft sciences. To do a hypothesis test, set up two contradictory statements which are the hypotheses. The first hypothesis is often the accepted belief or is assumed to be true. Then conduct the test to determine if the data supports or does not support the first hypothesis.

There are basic steps to a hypothesis test:

Step 1: Formulate the 2 hypotheses. They are called the null hypothesis and the alternate hypothesis.

• H_o is the first hypothesis and is called the null hypothesis.
• H_a is the second hypothesis and is called the alternate hypothesis. It is contradictory to the null hypothesis.

Step 2: Determine the random variable and the distribution for the test. Knowing the random variable and the distribution for the test, calculate a statistic from the data (for example, the sample mean or the estimated proportion) that will assess the evidence against the null hypothesis.

Step 3: Using the statistic, calculate the p-value. The p-value is the probability that the statistic calculated from the data will happen purely by chance when the null hypothesis is true. A smaller p-value indicates stronger evidence against H_o.

Step 4: Make a comparison of the p-value with a fixed or pre-conceived significance level, α. α acts as a cut-off point below which we agree that the statistic calculated from the data is statistically significant.

Then make a decision:

• If α> p-value, then we reject the null hypothesis.
• If α< p-value, then we do not reject the null hypothesis.
• If α = p-value, then our test is inconclusive. We, most likely, would gather more data and run at least one more test.

Step 5: Write an appropriate conclusion to the hypothesis test so that everyone can understand the result.

If we reject the null hypothesis, we write the alternate hypothesis in a sentence as the conclusion.

If we do not reject the null hypothesis, we simply write it in a sentence as the conclusion.

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The Null and Alternate Hypotheses and P-value

The null hypothesis, H_o, if written as a mathematical statement, has one of the following symbols in it:

(equal symbol, less than or equal symbol, greater than or equal symbol)

The alternate hypothesis, H_a, if written as a mathematical statement, has one of the following symbols in it:

(not equal symbol, less than symbol, greater than symbol)

Remember, H_o and H_a are contradictory.

The following examples demonstrate either a test of a single population mean, μ, or a test of a single population proportion, p. A test is either left-tailed, right-tailed, or two-tailed. The shaded area in the graphs show the p-value. Example:

Example:

Example:

Left-tailed test.

The "<"

in H_a tells us this fact.

Example:

Two-tailed test. The

""

in H_a tells us this fact.

Notice that the p-value is divided equally in both tails.

NOTE: We use technology (TI-83 or TI-84 calculators) to calculate the p-value for hypothesis tests in the next three sections of this Lesson. For this Lesson, we do hypothesis testing for a single population mean or a single population proportion.

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Pre-conceived Level of Signigicance, α

Fixed α is also known as a pre-conceived α. It is the probability to which we compare the p-value. A fixed α level of 0.05 or 0.01 is most commonly used. These levels were chosen before we had computers and had to use limited tables. Still, today, we often use 0.05 or 0.01. However, many hypothesis testers choose other values. For example, a medical hypothesis test might have an α of 0.001. A hypothesis test that was concerned with racial bias in jury selection in the years between 1960 and 1980 used an α equivalent to the probability of getting 3 consecutive royal flushes in poker! (This probability is almost 0.)

For this class, if α is not given, we will use 0.05.

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Think About It

Do the Try-It examples in Introductory Statistics. You can check some of the answers in the back of the book. Trying the examples will help you understand how to set up hypothesis.

Please continue to the next section of this lesson.

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Up » 9.1 Hypothesis Testing » 9.2 Hypothesis Testing - Known » 9.3 Hypothesis Testing- Unknown » 9.4 Hypothesis Testing for a Single Population Proportion » 9.5 Type I and II Errors

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