Exam 11: B: The Analysis of Variance

Product Markup Narrative A chain of jewellery stores conducted an experiment to investigate the relationship between price and location and the demand for its diamonds. Six small-town stores were selected for the study, as well as six stores located in large suburban malls. Two stores in each of these locations were assigned to each of three item percentage markups. The percentage gain (or loss) in sales for each store was recorded at the end of one month. The data are shown in the accompanying table. Markup -Refer to Product Markup Narrative. Find a 95% confidence interval for the difference in mean change in sales for stores in small towns versus those in suburban malls if the stores are using price markup 3.

Absenteeism Narrative The data shown below are collected using randomized design. The data values represent the number of days absent from work for three independent samples of workers. -Refer to Absenteeism Narrative. Calculate SST and MST.

Fuel Consumption Study Narrative A study was conducted to compare fuel consumption of medium-size trucks for three brands of gasoline, A, B, and C. Four trucks of the same make and model were used in the experiment, and each gasoline brand was tested in each truck. Using each brand in the same truck has the effect of eliminating (blocking out) truck-to-truck variability. The data (litres per 100 km) are as follow: Automobile -Refer to Fuel Consumption Study Narrative. Do the data provide sufficient evidence to indicate a difference in mean fuel consumption for the three brands of gasoline? Explain.

Healthy Dog Food Narrative A premium dog food manufacturer claims that, since its food is so highly nutritious, you won't need to feed your dog as much as with other dog foods. The premium dog food was compared to three other brands with respect to how much a dog needed to eat, in kilograms per week, to maintain its current weight and for the dog to remain healthy. A random sample of 24 similar dogs was selected and divided into 4 groups (i.e., 6 dogs were randomly assigned to each dog food brand). The following partial analysis table was computed: Analysis of Variance for Dog Food Brands -Refer to Healthy Dog Food Narrative. What conclusion can be drawn? Explain.

Wrapping Colour Narrative A firm's product can be wrapped in any of three colours: red, white, and black. The manager wants to test whether mean monthly sales (in $1000s) are the same, regardless of the colour. Viewing the past five months as a random sample, the manager collected the data shown below. Sales History (thousands of dollars) -Refer to Wrapping Colour Narrative. State the appropriate null and alternative hypotheses.

Food Irradiation Narrative In recent years the irradiation of food to reduce bacteria and preserve the food longer has become more common. A company that performs this service has developed four different methods of irradiating food. To determine which is best, it conducts an experiment where different foods are irradiated and the bacteria count is measured. As part of the experiment, the following foods are irradiated: beef, chicken, turkey, eggs, and milk. The results are shown below. Bacteria Count -Refer to Food Irradiation Narrative. Can the company infer at the 1% significance level that differences in the bacteria count exist among the four irradiation methods?

Breaking Strength of Thread Narrative A textile company is interested in knowing if there is a difference in the breaking strength of four different kinds of thread. = 6605.02, and = 362.2 -Refer to Breaking Strength of Thread Narrative. Calculate the test statistic.

Healthy Dog Food Narrative A premium dog food manufacturer claims that, since its food is so highly nutritious, you won't need to feed your dog as much as with other dog foods. The premium dog food was compared to three other brands with respect to how much a dog needed to eat, in kilograms per week, to maintain its current weight and for the dog to remain healthy. A random sample of 24 similar dogs was selected and divided into 4 groups (i.e., 6 dogs were randomly assigned to each dog food brand). The following partial analysis table was computed: Analysis of Variance for Dog Food Brands -Refer to Healthy Dog Food Narrative. What is the value of the test statistic?

Randomized Block Design Narrative The partially completed ANOVA table for a randomized block design is presented below: -Refer to Randomized Block Design Narrative. Do the data present sufficient evidence to indicate differences among the treatment means? Provide a justification for your answer. (Test using = 0.05.)

Travel Agency Narrative A travel agency primarily reserves flights with four major airlines. The agency would like to know if the true mean price for the four airlines is the same. Below are the prices for flights leaving from the same city and travelling to five different destinations. -Refer to Travel Agency Narrative. Perform an analysis of variance treating the destinations as blocks. Use $\alpha$ = 0.05.

Fill in the blanks (identified by asterisks) in the following partial ANOVA table:

Baseball Controversy Narrative In recent years a controversy has arisen in major league baseball. Some players have been accused of "doctoring" their bats to increase the distance the ball travels. However, a physics professor claims that the effect of doctoring is negligible. A major league manager decides to test the professor's claim. He doctors two bats by inserting cork into one and rubber into another. He then tells five players on his team to hit a ball with an undoctored bat and with the doctored bats. The distances are measured and listed below. Distance Ball Travels (in feet) -Refer to Baseball Controversy Narrative. Do these data provide sufficient evidence with the 5% level of significance to refute the professor's claim? Justify your conclusion.

Laundry Detergent Preference Narrative A consumer would like to know if there is a difference in the performance of three different laundry detergents. Eighteen pieces of white cloth were soiled with grape juice then washed in one of the three detergents A, B, or C. The resulting whiteness readings are listed below, where the larger number indicates whiter fabric. -Refer to Laundry Detergent Preference Narrative. Calculate the value of the test statistic F.

Running Shoes Narrative An avid runner was interested in whether there is a significant difference in the average wear (measured in weeks of use) among three brands of running shoes. To answer the question, the runner randomly selected six runners and assigned them to wear each of the three brands of running shoes until the shoes wore out. Each of the runners wore the brands of shoes in a random order. After the data had been recorded, the following output was generated using statistical software: Two-way Analysis of Variance -Refer to Running Shoes Narrative. What are the blocks?

Insurance Company Narrative An insurance company is considering opening a new branch in Lethbridge. The company will choose the final location from two locations within the city. One of the factors in the decision is the annual family income (in thousands of dollars) of five families randomly sampled from a radius of five miles from the potential locations. -Refer to Insurance Company Narrative. What is the mathematical relationship between the observed t and observed F-test statistics from the previous two questions? Does the same relation hold true for the corresponding critical values? Why or why not?

Laptop Battery Charge Time Narrative A computer laboratory manager must choose between three brands of battery packs for use in a laboratory of laptop computers. A major concern is the time, in hours, the battery packs will function before needing to be recharged. The manager obtains a random sample of eight observations for each of the three brands and records the following information: = 1150.72, and = 165.6 -Refer to Laptop Battery Charge Time Narrative. Find and interpret the approximate p-value.

Statistical Software Narrative A professor of statistics is trying to determine which of three statistical software programs is best for his students. He believes that the time (in hours) it takes a student to master particular software may be influenced by gender. A 3 * 2 factorial experiment with three replicates was designed, as shown below: Gender -Refer to Statistical Software Narrative. Test at the 10% significance level to determine if differences exist among male and female students.

Physicians depend on laboratory test results when managing medical s such as diabetes or epilepsy. In a uniformity test for glucose tolerance, three different laboratories each sent = 5 identical blood samples from a person who had drunk 50 milligrams (mg) of glucose dissolved in water. The laboratory results (in mg/dL) are listed here: Use Tukey's method for paired comparisons to rank the three treatment means. Use = 0.05.

Incentive Pay Narrative A company conducted an experiment to determine the effect of two types of incentive pay plans on worker productivity for workers of two shifts. The company used an equal number of production workers from each of the two shifts and one-half of these workers were assigned to each plan. Then five workers from each pay plan-shift combination were selected and their productivity (in number of items produced) recorded for a one-week period. The following output was generated using statistical software: Two-way analysis of variance -The data that follow are observations collected from an experiment that compared four treatments, A, B, C, and D, within each of three blocks, using a randomized block design. Treatment a. Do the data present sufficient evidence to indicate differences among the treatment means? Test using = 0.05. b. Do the data present sufficient evidence to indicate differences among the block means? Test using = 0.05. c. Find a 95% confidence interval for the difference in means for treatments A and B. d. Does it appear that the use of a randomized block design for this experiment was justified? Explain.

Headache Treatments Narrative The following data were generated from a 2 * 2 factorial experiment with three replicates, where factor A levels represent two different injection procedures of an anesthetic to the occipital nerve (located in the back of the neck), and factor B levels represent two different drugs that physicians recommend to increase the effectiveness of the injections. Three headache patients were randomly selected for each combination of injection and drug. Factor B -Refer to Headache Treatments Narrative. Test at the 5% significance level to determine if factors A and B interact.