The Null and Alternative Hypotheses and Errors in Hypothesis Testing

· From the textbook, Business Statistics in Practice, read the following chapters:

· Hypothesis Testing

· Decision Theory

 

CHAPTER 9: Hypothesis Testing

Chapter Outline

9.1 The Null and Alternative Hypotheses and Errors in Hypothesis Testing

9.2 z Tests about a Population Mean: σ Known

9.3 t Tests about a Population Mean: σ Unknown

9.4 z Tests about a Population Proportion

9.5 Type II Error Probabilities and Sample Size Determination (Optional)

9.6 The Chi-Square Distribution (Optional)

9.7 Statistical Inference for a Population Variance (Optional)

Hypothesis testing is a statistical procedure used to provide evidence in favor of some statement (called a hypothesis). For instance, hypothesis testing might be used to assess whether a population parameter, such as a population mean, differs from a specified standard or previous value. In this chapter we discuss testing hypotheses about population means, proportions, and variances.

In order to illustrate how hypothesis testing works, we revisit several cases introduced in previous chapters and also introduce some new cases:

The Payment Time Case: The consulting firm uses hypothesis testing to provide strong evidence that the new electronic billing system has reduced the mean payment time by more than 50 percent.

The Cheese Spread Case: The cheese spread producer uses hypothesis testing to supply extremely strong evidence that fewer than 10 percent of all current purchasers would stop buying the cheese spread if the new spout were used.

The Electronic Article Surveillance Case: A company that sells and installs EAS systems claims that at most 5 percent of all consumers would never shop in a store again if the store subjected them to a false EAS alarm. A store considering the purchase of such a system uses hypothesis testing to provide extremely strong evidence that this claim is not true.

The Trash Bag Case: A marketer of trash bags uses hypothesis testing to support its claim that the mean breaking strength of its new trash bag is greater than 50 pounds. As a result, a television network approves use of this claim in a commercial.

The Valentine’s Day Chocolate Case: A candy company projects that this year’s sales of its special valentine box of assorted chocolates will be 10 percent higher than last year. The candy company uses hypothesis testing to assess whether it is reasonable to plan for a 10 percent increase in sales of the valentine box.

 9.1: The Null and Alternative Hypotheses and Errors in Hypothesis Testing

One of the authors’ former students is employed by a major television network in the standards and practices division. One of the division’s responsibilities is to reduce the chances that advertisers will make false claims in commercials run on the network. Our former student reports that the network uses a statistical methodology called hypothesis testing to do this.

Chapter 9

To see how this might be done, suppose that a company wishes to advertise a claim, and suppose that the network has reason to doubt that this claim is true. The network assumes for the sake of argument that the claim is not valid. This assumption is called the null hypothesis. The statement that the claim is valid is called the alternative, or research, hypothesis. The network will run the commercial only if the company making the claim provides sufficient sample evidence to reject the null hypothesis that the claim is not valid in favor of the alternative hypothesis that the claim is valid. Explaining the exact meaning of sufficient sample evidence is quite involved and will be discussed in the next section.

The Null Hypothesis and the Alternative Hypothesis

In hypothesis testing:

1 The null hypothesis, denoted H0, is the statement being tested. Usually this statement represents the status quo and is not rejected unless there is convincing sample evidence that it is false.

2 The alternative, or research, hypothesis, de noted Ha, is a statement that will be accepted only if there is convincing sample evidence that it is true.

Setting up the null and alternative hypotheses in a practical situation can be tricky. In some situations there is a condition for which we need to attempt to find supportive evidence. We then formulate (1) the alternative hypothesis to be the statement that this condition exists and (2) the null hypothesis to be the statement that this condition does not exist. To illustrate this, we consider the following case studies.

 EXAMPLE 9.1: The Trash Bag Case1

A leading manufacturer of trash bags produces the strongest trash bags on the market. The company has developed a new 30-gallon bag using a specially formulated plastic that is stronger and more biodegradable than other plastics. This plastic’s increased strength allows the bag’s thickness to be reduced, and the resulting cost savings will enable the company to lower its bag price by 25 percent. The company also believes the new bag is stronger than its current 30-gallon bag.

The manufacturer wants to advertise the new bag on a major television network. In addition to promoting its price reduction, the company also wants to claim the new bag is better for the environment and stronger than its current bag. The network is convinced of the bag’s environmental advantages on scientific grounds. However, the network questions the company’s claim of increased strength and requires statistical evidence to justify this claim. Although there are various measures of bag strength, the manufacturer and the network agree to employ “breaking strength.” A bag’s breaking strength is the amount of a representative trash mix (in pounds) that, when loaded into a bag suspended in the air, will cause the bag to rip or tear. Tests show that the current bag has a mean breaking strength that is very close to (but does not exceed) 50 pounds. The new bag’s mean breaking strength μ is unknown and in question. The alternative hypothesis Ha is the statement for which we wish to find supportive evidence. Because we hope the new bags are stronger than the current bags, Ha says that μ is greater than 50. The null hypothesis states that Ha is false. Therefore, H0 says that μ is less than or equal to 50. We summarize these hypotheses by stating that we are testing

H0: μ ≤ 50   versus   Haμ > 50

The network will run the manufacturer’s commercial if a random sample of n new bags provides sufficient evidence to reject H0: μ ≤ 50   in favor of   Haμ > 50.

 EXAMPLE 9.2: The Payment Time Case

Recall that a management consulting firm has installed a new computer-based, electronic billing system for a Hamilton, Ohio, trucking company. Because of the system’s advantages, and because the trucking company’s clients are receptive to using this system, the management consulting firm believes that the new system will reduce the mean bill payment time by more than 50 percent. The mean payment time using the old billing system was approximately equal to, but no less than, 39 days. Therefore, if μ denotes the mean payment time using the new system, the consulting firm believes that μ will be less than 19.5 days. Because it is hoped that the new billing system reduces mean payment time, we formulate the alternative hypothesis as Haμ < 19.5 and the null hypothesis as H0: μ ≥ 19.5. The consulting firm will randomly select a sample of n invoices and determine if their payment times provide sufficient evidence to reject H0: μ ≥ 19.5 in favor of Haμ < 19.5. If such evidence exists, the consulting firm will conclude that the new electronic billing system has reduced the Hamilton trucking company’s mean bill payment time by more than 50 percent. This conclusion will be used to help demonstrate the benefits of the new billing system both to the Hamilton company and to other trucking companies that are considering using such a system.

 EXAMPLE 9.3: The Valentine’s Day Chocolate Case 2

A candy company annually markets a special 18 ounce box of assorted chocolates to large retail stores for Valentine’s Day. This year the candy company has designed an extremely attractive new valentine box and will fill the box with an especially appealing assortment or chocolates. For this reason, the candy company subjectively projects—based on past experience and knowledge of the candy market—that sales of its valentine box will be 10 percent higher than last year. However, since the candy company must decide how many valentine boxes to produce, the company needs to assess whether it is reasonable to plan for a 10 percent increase in sales.

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