Chapter 9 Fundamentals of Hypothesis Testing: One-Sample Tests

9.1: Hypothesis Testing Methodology Confidence Intervals were our first Inference Hypothesis Tests are our second Inference “Methodology” implies a series of steps: 1. Develop hypotheses 2. Determine decision rule 3. Calculate test statistic 4. Compare results from 2 and 3: make a decision 5. Write conclusion

Step 1: Develop hypotheses You will need to develop 2 hypotheses: 1.Null hypothesis 2.Alternative hypothesis –Hypotheses concern the population parameter in question (ie “µ” or “π” or other)

The Null Hypothesis A theory or idea about the population parameter. Always contains some sort of equality. Very often described as the hypothesis of “no difference” or “status quo.” H 0 : µ = 368

The Alternative Hypothesis An idea about a population parameter that is the opposite of the idea in the null hypothesis NEVER contains any sort of equality! H 1 : µ  368 (sometimes use H a :)

Hypotheses Null and alternative hypotheses are mutually exclusive and collectively exhaustive. Our sample either contains enough information to reject the NULL hypothesis OR the sample does not contain enough information to reject the null hypothesis.

“Proof” There is no proof. There is only supporting information.

Step 2: Decision Rule or “Rejection Region” Always says something like “we shall reject H 0 for some extreme value of the test statistic.” The “Rejection Region” is the range of the test statistic that is extreme enough—so extreme that the test statistic probably would not occur IF the null hypothesis is true. Figure 9.1 shows the rejection region for a hypothesis test of the mean. The “critical value” is looked up based on the error rate that you are comfortable with.

Step 3: Calculating the Test Statistic The test statistic depends on the sampling distribution in use. This depends on the parameter. This will be determined the same way it was in chapter 8.

Step 4 & 5: Decision and Conclusion The decision is always either (1) reject H 0 or (2) fail to reject H 0. This is determined by evaluating the decision rule in step 2. The conclusion always says “At α = 0.05, there is (in)sufficient information to say H 1 ”

Alpha α is the probability of committing a Type I error: erroneously rejecting a true Null Hypothesis. α is called “The Level of Significance” α is determined before the sample results are examined. α determines the critical value and rejection region(s). α is set at an acceptably low level.

Beta Beta is the probability of committing a Type II error: erroneously failing to reject a false null hypothesis. Beta depends on several factors and it cannot be arbitrarily set. Beta can be indirectly influenced.

Compliments of Alpha and Beta (1-α) is called the confidence coefficient. This is what we used in Chapter 8. (1-beta) is called the Power of the test. Power is the chance of rejecting a null hypothesis that ought to be rejected, ie a false null. Bigger is better. Power cannot be set directly.

9.2: z Test of Hypothesis for the mean Use this test ONLY for the mean and only when σ is known. There are two approaches: –critical value approach –“p” value approach

Critical Value Approach Remember your methodology (steps): 1.Create hypotheses 2.Create decision rule –depends on α –depends on distribution 3.Calculate test statistic 4.State the result 5.State the managerial conclusion

Hypotheses The discussion in 7.2 assumes a two- tail test because the sample mean might be extremely large or extremely small. –Either one would make you think the null hypothesis is wrong.

Rejection Region The standard approach requires that the value of α be divided evenly between the tail areas. These tail areas are called the “rejection region.”

Conclusion See Step 6 on pages

“p-value” approach Rewrite the decision rule to say, “we will reject the null hypothesis if the ‘p-value’ is less than the value of α.” “p-value” definition, page 309. “p-value” is called the observed level of significance. Excel--most statistical software--does a good job of this (that’s why it’s a popular approach).

Estimation and Hypothesis Testing The two inferences are closely related. Estimation answers the question “what is it?” Hypothesis Testing answers the question “is it ______ than some number?” See page 312.

9.3: One-Tail Tests The rejection region is one single area. Sometimes called a directional test. Mechanics: –see the text example Problem identification: –hypothesis test or interval estimation? –One-tail or Two-tail? –If One-tail, which is the null?

Text Example Page 314, the “milk problem.” Are we buying “watered-down milk” ? –Watered-down milk freezes at a colder temperature than normal milk. –What are the null and alternative hypotheses? –Hint: what do you want to conclude? –Hint: what is the hypothesis of action? –Hint: what is the hypothesis of status quo?

Mechanics of the One-tailed test Different hypotheses. Different decision rule/rejection region. Different “p-value” or observed significance of observed level of significance.

Consider Problem 9.44, page 317 Reading only the context, not the steps (a, b, etc.), can you tell that the problem calls for a hypothesis test? –Knowing that a test of hypothesis is called for, can you determine that a one-tail test is appropriate? Knowing that a one-tail test is to be used, can you set up the hypotheses?

9.4: t test of Hypothesis for the Mean (σ Unknown) When σ is unknown, the distribution for x-bar is a “t” distribution with n-1 degrees of freedom. Use sample standard deviation “s” to estimate σ. This test is more commonly used than the z test.

Assumptions Random Sample. You must assume that the underlying population, i.e. the underlying random variable x is distributed normally. This test is very “robust” in that it does not lose power for small violations of the above assumption.

Methodology You still need your 5 step Hypothesis Test Methodology. –The critical value approach is the same as that for the z test. –The p-value method does not work as well when done by hand because of limitations in the “t” table. –One- and Two-tail tests are possible. You could add another step to check assumptions.

EXAMPLE 9.54, page 323

9.5: z Test of Hypothesis for the Proportion For the nominal variable—variable values are categories and you tend to describe the data set in terms of proportions. Both one- and two-tail tests are possible. Problem 9.72 on page 329 is a good example.

Assumptions The number of observations of interest (successes) and the number of uninteresting observations (failures) are both at least 5.