Chapter 10: Hypothesis Testing Using a Single Sample

Section 10.1: Hypotheses and Test Procedures

• Hypotheses – a claim or statement about either the value of a single population characteristic or the values of several population characteristics.

• Test of hypotheses or test procedures – a method for using sample data to decide between two competing claims about a population characteristic.

• Null Hypotheses – denoted by H0, is a claim about a population characteristic that is initially assumed to be true.

• Alternate Hypotheses – denoted by Ha, is the competing claim.

Example

• Because of variation in the manufacturing process, tennis balls produced by a particular machine do not have identical diameters. Let μ denote the true average diameter for tennis balls currently being produced. Suppose that the machine was initially calibrated to achieve the design specification μ = 3 inches. However, the manufacturer is now concerned that the diameters no longer conform to this specification.

• That is μ ≠ 3 inches must now be considered as a possibility. If sample evidence suggests that μ ≠ 3 inches, the production process will have to be halted while the machine is recalibrated. Because this production break is costly, the manufacturer wants to be quite sure that μ ≠ 3 inches before undertaking recalibration.

• Under these circumstances, a sensible choice of hypotheses is

H0: μ = 3 (the specification is being met, so recalibration is unnecessary)

Ha: μ ≠ 3 (the specification is not being met, so recalibration is necessary)

Only compelling sample evidence would then result in H0 being rejected in favor of Ha.

• The form of a null hypotheses is

H0: population characteristic = hypothesized value

Where the hypothesized value is a specific number determined by the problem context

The alternative hypothesis has one of the following three forms:

Ha: population characteristic > hypothesized valueHa: population characteristic < hypothesized valueHa: population characteristic ≠ hypothesized value

Example

• A medical research team has been given the task of evaluating a new laser treatment for certain types of tumors. Consider the following two scenarios:

• Scenario 1: The current standard treatment is considered reasonable and safe by the medical community, has no major side effects, and has a known success rate of 0.85 (85%).

• Scenario 2: The current standard treatment sometimes has serious side effects, is costly, and has a know success rate of .30 (30%).

• In the first scenario, research efforts would probably be directed toward determining whether the new treatment has a higher success rate than the standard treatment. Unless convincing evidence of this is presented, it is unlikely that current medical practice would be changed. With π representing the true proportion of successes for the laser treatment, the following hypotheses would be tested:

H0: π = .85 verses Ha: π > .85

• In this case, rejection of the null hypothesis is indicative of compelling evidence that the success rate is higher for the new treatment.

• In the second scenario, the current standard treatment does not have much to recommend it. The new laser treatment may be considered preferable because of cost or because it has fewer or less serious side effects, as long as the success rate for the new procedure is no worse than that of the standard treatment. Here, researchers might decide to test the hypotheses

H0: π = .30 versus Ha: π < .30

• If the null hypotheses is rejected, the new treatment will not be put forward as an alternative to the standard treatment, because there is strong evidence that the laser method has a lower success rate.