Chapter 10 Inference Procedures and Type I, Type II Error Classroom Ex

Chapter 10 Inference Procedures and Type I, Type II Error Classroom Ex.

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1. This Wine Stinks Sulfur compounds cause “off-odors” in wine, so winemakers want to know the odor threshold, the lowest concentration of a compound that the human nose can detect. The odor threshold for dimethyl sulfide (DMS) in trained wine tasters is about 25 micrograms per liter of wine (µg/l). The untrained noses of consumers may be less sensitive, however. Here are the DMS odor thresholds for 10 untrained students.

31 31 43 36 23 34 32 30 20 24

Assume that the standard deviation of the odor threshold for untrained noses is known to be σ = 7 µg/l. a) Make a stemplot to verify that the distribution of the sample is roughly symmetric with no outliers. This will give you some faith that it was drawn from a reasonably normal population. (More data would confirm that there are no meaningful departures from normality).

b) Give a 95% confidence interval for the mean DMA odor threshold among all students. (Toolbox!!) d) Are you convinced that the mean odor threshold for students is higher than the published threshold, 25 µg/l? Carry out a significance test to justify your answer. (Another toolbox  . If this toolbox has elements in common with part b), you may say, “See above.”

2. Explaining Statistical Significance When asked to explain the meaning of “the P-value was P = .03,” a student says, “This means there only a probability of .03 that the null hypothesis is true.” a) Is this an essentially correct explanation? Explain your answer.

b) Another student, when asked why statistical significance appears so often in research reports, says, “Because saying that results are significant tells us that they cannot easily be explained by chance alone.” Do you think that this statement is essentially correct?

c) What might you add to the statement in Part b) to make it even better? 3. Opening a Restaurant You are thinking about opening a restaurant and are searching for a good location. From research you have done, you know that the mean income of those living near the restaurant must be over $45,000 to support the type of upscale restaurant you wish to open. You decide to take a simple random sample of 50 people living near one potential location. Based on the mean income of this sample, you will decide whether to open a restaurant there. A number of similar studies have shown that σ = $5000. a) Describe the two types of errors that you might make. Identify which is a Type 1 error and which is a

Type 2 error.

b) Which of the two types of error is most serious. Explain! c) State the null and alternate hypotheses. d) If you had to choose one of the “standard” significance levels for your significance test, would you chose α = .01, .05, or .10? Justify your choice. e) Based on your choice in part d), how high will the sample mean need to be before you decide to open a restaurant in that area? (Hint: This is finding x-bar *--the critical value for the test—the line in the sand beyond which you would consider a sample average income to be sufficiently high to justify opening the restaurant. f) You take a sample of 50 households and find the mean to be $46200. Should you open the restaurant? Using all of your stats superpowers, justifying your decision, such as running a hypothesis test.

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