Hypothesis testing tells us whether the weight of statistical evidence falls for
ID: 1720597 • Letter: H
Question
Hypothesis testing tells us whether the weight of statistical evidence falls for or against a certain contention. We divide the world into two possibilities, a null hypothesis and an alternative hypothesis. The null hypothesis is sometimes referred to as the dull hypothesis because it almost always represents the status quo. For example, it represents the cure rate of existing drugs, the penetration rate of existing sales programs, or student test scores under the existing curriculum design. The alternative hypothesis challenges the null hypothesis. For example, it tests to see if a new drug has a superior cure rate compared to the old drug, whether a new sales program has increased market penetration, or whether a new curriculum design is improving student test scores. The weight of statistical evidence must be relatively large before we reject the null hypothesis and accept the alternative hypothesis. Statisticians are never sure of their findings, because chance plays a large part in the sampling process. Sometimes samples do not represent the population from which they were drawn, strictly because the sample members were not indicative. Say we were studying household income and by chance our random sample included even a few of the following names; Bill Gates, Warren Buffet, Larry Ellison, Tom Galisano, Tiger Woods, Martha Stewart, Oprah Winfrey, or Carl Ichan. Our sample would over estimate average household income because the members of the sample, by chance, were outliers. (In a small sample having even one of these wealthy people would skew our results – recall the discussion about measures of central tendency.) Because we cannot examine the whole population, we must allow for the uncertainty the sampling process introduces. The confidence level indicates how sure we want to be about our conclusions. If we test at a 95% confidence level and reject the null hypothesis, we can be 95% sure that the alternative hypothesis is true. Consider the following example. Our existing promotion program produces a mean penetration of 25% in our regular markets. A new promotion program is tried in a few of our reliable test markets. It produces a mean sales penetration of 27% in those markets. Is this enough of a difference that we believe the new program is superior? Or is this possibly a chance event that won’t replicate to all markets if we adopt the new program nationwide? Null Hypothesis: Ho: Mean of the New Sales Program is less than or equal to 25% Alternative Hypothesis (Ha or H1) Ha: Mean of the New Sales Program is greater than 25% Or, in statistical notation: H0: 25% (Sometimes, this can be written as = 25% Ha: > 25% as the default null hypothesis – Bluman Text 4e,5e) © 20072010 Gianturco, Teitelbaum Page 2 of 4 Statistics: Case Study #4 Spring 2010 Say the statistical tests we use tell us to reject the null hypothesis. At the 95% level of confidence, we are 95% sure that the result we see is real, that if we adopt the new sales program nationwide, we will see superior sales penetration (greater than 25%) in our markets. If by chance our test markets were not indicative of the national markets, or if some other chance element entered into our data, we might find that when we applied the new program nationwide, sales might not improve over 25%. In fact, sale might fall. Put another way, since we are testing at the 95% level of confidence, there is also a 5% chance that we are wrong. You might ask, why not test at the 99% level then? If we do, then there is only a 1% chance that we reach the wrong conclusion (this is referred to as in this case =0.01). We could, but the higher we set the level of confidence, the harder it is to reject the null hypothesis. For example, if the test markets produce a result of 27% mean penetration, we might be 95% sure the result is real (not a result of chance), but we might not be 99% sure. Maybe we need to have a 30% penetration in the test markets before we are 99% sure the result is real. We might reject a good sales program if we make the criteria too strict. Setting the bar too high has consequences. In pharmaceutical research the bar is set very high. Why? We don’t want to release inferior drugs that can harm the public. So 95% certainty (confidence level) is too low for most medical research. We would rather leave a good new drug on shelf than take a chance that a new drug is inferior (or does harm), so we set the confidence level at 99% or higher. A Type I Error occurs when we reject a null hypothesis that is in fact true. In our previous example, we conclude that the 27% test market penetration is sufficient to prove the viability of the new sales program. However, when we release the program nationwide, sales are generally no better (and possibly worse). A Type II Error occurs when we do not reject a null hypothesis that is in fact false. In our previous example, we conclude that the 27% test market penetration is insufficient to prove the viability of the new sales program. We don’t run the new program when in fact it would have improved sales – we forgo profits. In this portfolio problem you are asked to design some experiments related to a hypothetical business expansion project. You are not asked to conduct the study or do any computations. You will sketch three problems. For each problem you will formalize a hypothesis (H0, null and Ha, alternative), and state the specific business ramifications of both Type I and Type II Errors. Use the Hypothesis Test Chart included in this PDF to help you determine which paradigm this case follows. Use proper notation. © 20072010 Gianturco, Teitelbaum Page 3 of 4 Statistics: Case Study #4 Spring 2010 Background A restaurant chain specializing in ice cream deserts contacted you to characterize the market for new franchise locations in Upstate NY. The firm is located in the Southwestern US, and has excellent experience there. However, they are opening a few test locations in Buffalo, Syracuse and Albany in the very near future, and have hired you, an Upstate NY statistical consultant, to collect and analyze the results. These restaurants follow a standard blueprint for design, operation, and location. They run a 2,150 sq foot facility in wealthier suburban areas, and usually locate in new plazas near other specialty restaurants that tend not to compete with them (Starbucks, Panera, Fridays, and similar restaurants.) Existing Southwestern locations take in an average of $82,000 a week in sales, with a highly reliable standard deviation of $5,200. Before deciding on their regional distribution strategy, the firm is opening three test restaurants, one in each city. They want you to examine the weekly revenue data and characterize the regional market for their restaurants. Analysis For each of the following three situations, show an experimental design, which includes, at a minimum: a) Indicate whether this is a one-tail or two-tail test, and why. Sketch the problem. b) State your hypotheses c) Identify the level of significance (with justification) d) State the ramifications of the two types of errors, specific to this business. Due to the statistical format and sketching, you may wish to neatly write out your answers to (a) and (b) and scan the resultant pages. Answers to parts (c) and (d) should be typed in a Word document. 1) Your original contact at the firm was the Senior Vice President (Sr. VP) for business development. She has never been to the Upstate New York, and was skeptical the new restaurants would do well compared to the Southwestern locations. After all the weather was hotter in the Southwest, winters were shorter, and Southwestern cities tended to have better economies overall. If the regional expansion went forward based on your analysis, and then didn’t pan out, the ice cream company would lose a major portion of its investment. The senior vice president was not pushing for this project, but rather was resisting a push from the CEO to move forward. The senior vice president was so sure the new stores would be inferior that he asked you to gather data from the new stores in the test markets, and statistically test for a pattern of inferior sales as compared to the Southern store base. The vice president confided in you that if your test proved him right, he could shelve the project now before the second phase of investment was committed. 2) You decided to buy your friend an ice cream for lunch at your favorite upscale ice cream parlor about ten miles away from the office. It was wintertime, and you noticed for the first time just how many local ice cream restaurants and stands there were in your area. Funny, they were almost all closed during the winter months. Despite the cold weather you had to wait a couple minutes for a table to clear at the parlor. You mention to an © 20072010 Gianturco, Teitelbaum Page 4 of 4 Statistics: Case Study #4 Spring 2010 employee that you must have hit a busy day, but they respond that it’s not unusually busy today. You start to think about the conversation you had earlier with the Sr. VP, and you wonder if he or you, or anyone at your client firm really understands the local market at all. You are not really sure if sales locally will be any different than in Southwestern markets. Show a design that reflects this line of thought. 3) When you get back to the office you get a call from the CEO of the ice cream company. He wanted to clarify a few things for you. He surprises you by indicating that the firm faced a rather large downside (it would have a material effect on the bottom-line earnings) if the new facilities are understaffed when they open. It takes a long time to train employees for a new opening, and training is very expensive. However, if new customers have a bad experience at a new store they don’t come back, and their friends never come in. The word usually spreads faster about a bad experience than a good one. The CEO related that the only closings the firm ever had were when they had a poor start in a new and otherwise attractive location. Proper staffing was directly related to volume. Understaffed new facilities based on poor planning presented a clear risk. The CEO was insistent that the firm not face this situation. Will the results from the test markets give clear evidence that the new stores would do better than the average Southern locations? Design an experiment that strictly examines this risk.
Explanation / Answer
is it question?
see before posting the question jest see it....
Related Questions
Navigate
Integrity-first tutoring: explanations and feedback only — we do not complete graded work. Learn more.