we began our study of statistical inference by describing how we could select a random sample and then use the sample values to estimate the value of a population parameter. Recall that a sample is a part or subset of the population, while a parameter is a value calculated from the entire population. In Chapter 9 we estimated a population parameter from a sample statistic. In addition, we developed a range of values, called a confidence interval, within which we expected the population value to be located.
In this chapter, rather than developing a range of values within which we expect the population parameter to occur, we will conduct a test of hypothesis regarding the validity of a statement about a population parameter.
Two statements called hypotheses are made regarding the possible values of population parameters.
What is a Hypothesis?
A hypothesis is a statement about a population.
Hypothesis: A statement about a population parameter subject to verification.
In statistical analysis we make a claim, that is, state a hypothesis, and then follow up with tests to verify the assertion or to determine that it is untrue.
What is Hypothesis Testing?
The terms hypothesis testing and testing a hypothesis are used interchangeably. Hypothesis testing starts with a statement about a population parameter such as the mean.
Hypothesis testing: A procedure based on sample evidence and probability theory to determine whether the hypothesis is a reasonable statement.
For example, one statement about the performance of a new model car is that the mean miles per gallon is 30. The other statement is that the mean miles per gallon is not 30. Only one of these statements is correct.
Five-Step Procedure for Testing a Hypothesis
Statistical hypothesis testing is a five-step procedure. These steps are:
Step 1
Step 2
Step 3
Step 4
Step 5
State null and alternate hypotheses
→
Select a level of significance.
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Identify the test statistic.
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Formulate a decision rule.
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Take a sample, arrive at decision.
Do not reject H0 or reject H0 and accept H1
When we arrive at Step 5, we are ready to either accept or reject the null hypothesis. You should be aware that hypothesis testing as used by statisticians does not provide proof that something is
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Park University - EC315 - Chapter 10 Lecture Notes Page 2 true in the manner that a mathematician proves a statement. However, in cases where the null hypothesis is rejected, it does provide “proof beyond a reasonable doubt” that the null hypothesis is not true. The steps involved in hypothesis testing will now be described in more detail.
First we will concentrate on testing a hypothesis about a population mean. Then we will consider hypothesis testing for a population proportion. For a mean:
Step 1. State the null hypothesis (H0) and the alternate hypothesis (H1).
The first step is to state the hypothesis being tested. It is called the null hypothesis, designated H0 , and read H sub zero. The capital letter H stands for hypothesis, and the subscript zero implies “no difference.”
Null hypothesis: A statement about the value of a population parameter.
For example, a recent newspaper report made the claim that the mean length of a hospital stay was 3.3 days. You think that the true length of stay is some other length than 3.3 days.
The null hypothesis is written H0: μ = 3.3, where H0 is an abbreviation of the null hypothesis. The null hypothesis will always contain the equal sign. It is the statement about the value of the population parameter, in this case the population mean. The null hypothesis is established for the purpose of testing. On the basis of the sample evidence, it is either rejected or not rejected.
If the null hypothesis is rejected then we accept the alternate hypothesis.
Alternate hypothesis: A statement that is accepted if the sample data provide enough evidence that the null hypothesis is