### Statistics Prairie Flower Cereal, Inc

Statistics Prairie Flower Cereal, Inc

6. 83

Prairie Flower Cereal, Inc., is a small but growing producer of hot and ready-to-eat breakfast cereals. The company was started in 1910 by Gordon Thorson, a successful grain farmer. You have been asked to test the cereal-packing process of 18-ounce (510-gram) boxes of sugar-coated wheat cereal. Two machines are used for the packaging process. Twenty samples of five packages each are randomly sampled and weighed. The data are contained in the file Sugar Coated Wheat.

a. Compute the overall sample mean, sample variance, and variance of the sample means for each machine.

b. Determine the probability that a single sample mean is below 500 if the process is operating properly for each machine.

c. Determine the probability that a single sample mean is above 508 if the process is operating properly for each machine.

d. Using your statistical computer package, obtain 20 random samples of size n = 5 packages for each machine and compute the sample mean for each sample. Count the number of sample means that are below 500 and the number that are above 508.

6.84

Another product packaged by Prairie Flower Cereal, Inc., is an apple-cinnamon cereal. To test the packaging process of 40-ounce (1,134-gram) packages of this cereal, 23 samples of six packages each are randomly sampled and weighed. The lower and upper acceptance limits have been set at 1,120 grams and 1,150 grams, respectively. The data are contained in the data file Granola.

a. Compute the overall sample mean, sample variance, and variance of the sample means for each sample.

b. Compute the probability that the sample means will be within the acceptance limits.

c. Using your statistical computer package, obtain 23 random samples of size n = 6 and compute the sample mean for each sample. Count the number of sample means that are outside the acceptance limits.

7.90

What is the most common method to renew vehicle registration? In checking a random sample of 500 motor vehicle renewal registrations in one county, the finance department found that 200 were mailed, 160 were paid in person at the county finance department office, and the remainder was paid online at the county’s Web site. Phone registration renewals were not available.

a. Estimate the population proportion to pay for vehicle registration renewals in person at the county finance department office. Use a 90% confidence level.

b. Estimate the population proportion of online renewals. Use a 95% confidence level.

7.93

The county finance department in Exercise 7.90 also wants information about renewals of disabled parking placards. Suppose that in a sample of 350 transactions for disabled parking placards, it was found that 250 were paid electronically.

a. What is the margin of error for a 99% confidence interval estimate of the population proportion of disabled renewal transactions paid electronically?

b. Without calculating, is the margin of error for a 95% confidence interval estimate of the population proportion of disabled renewal transactions paid electronically larger, smaller, or the same as that found in part a for a 99% confidence interval?

8.34

The Healthy Eating Index measures on a 100-point scale the adequacy of consumption of vegetables, fruits, grains, milk, meat and beans, and liquid oils. This scale is called HEI2005 (Guenther et al. 2007). There are two interviews for each person in the study. The first interview is identified by daycode = 1 and the second interview is identified by daycode = 2. This data is stored in the data file HEI Cost Data Variable Subset. Find a 95% confidence interval estimate of the difference in the mean HEI–2005 scores between male and female participants at the time of their first interview.

9.59

In a random sample of 545 accountants engaged in preparing county operating budgets for use in planning and control, 117 indicated that estimates of cash flow were the most difficult element of the budget to derive.

a. Test at the 5% level the null hypothesis that at least 25% of all accountants find cash flow estimates the most difficult estimates to derive.

b. Based on the procedure used in part a, what is the probability that the null hypothesis would be rejected if the true percentage of those finding cash flow estimates most difficult was each of the following?

i. 20%

ii. 25%

iii. 30%

9.66

You have been hired by the National Nutrition Council to study nutrition practices in the United States. In particular they want to know if their nutrition guidelines are being met by people in the United States. These guidelines indicate that per capita consumption of fruits and vegetables should be more than 170 pounds per year, per capita consumption of snack foods should be less than 114 pounds, per capita consumption of soft drinks should be less than 65 gallons, and per capita consumption of meat should be more than 70 pounds. As part of your research you have developed the data file Food Nutrition Atlas, which contains a number of nutrition and population variables collected by county over all states. Variable descriptions are located in the chapter appendix. It is true that some counties do not report all the variables. Perform an analysis of the available data and prepare a short report indicating how well the nutrition guidelines are being met. Your conclusions should be supported by rigorous statistical analysis.

10.48

Joe Ortega is the product manager for Ole ice cream. You have been asked to determine if Ole ice cream has greater sales than Carl’s ice cream, which is a strong competitor. The data file Ole contains weekly sales and price data for the competing brands over the year in three different supermarket chains. These sample data represent a random sample of all ice cream sales for the two brands. The variable names clearly identify the variables.

a. Design and implement an analysis to determine if there is strong evidence to conclude that Ole ice cream has higher mean sales than Carl’s ice cream (α = 0.05). Explain your procedure and show all computations. You may include Minitab output if appropriate to support your analysis. Explain your conclusions.

b. Design and implement an analysis to determine if the prices charged for the two brands are different (α = 0.05). Carefully explain your analysis, show all computations, and interpret your results.

10.52

Previous research has suggested that immigrants in the United States have a stronger interest in good diet compared to the rest of the population. If true, this behavior could result from a desire for overall life improvement, historical experience from their previous country, or some other complex rationale. You have been asked to determine if immigrants (variable immigrant = 1) have healthier diets compared to nonimmigrants ( = 0). Perform an appropriate statistical test to determine if there is strong evidence to conclude that immigrants have better diets compared to natives.

You will do the analysis based first on the data from the first interview, create subsets of the data file using daycode = 1; then a second time, using data from the second interview, create subsets of the data file using daycode = 2. Note differences in the results between the first and second interviews.

11.77

Based on a sample of 30 observations, the population regression model

yi = β0 + β1xi + εi

was estimated. The least squares estimates obtained were as follows:

b0 = 10.1 and b1 = 8.4

The regression and error sums of squares were as follows:

SSR = 128 and SSE = 286

a. Find and interpret the coefficient of determination.

b. Test at the 10% significance level against a two-sided alternative the null hypothesis that b1 is 0.

c. Find

∑ 30 i = 1 ( x i − ¯ x ) 2

11.93

The federal nutrition guidelines prepared by the Center for Nutrition Policy and Promotion of the U.S. Department of Agriculture stress the importance of eating substantial servings of fruits and vegetables to obtain a healthy diet. You have been asked to determine if the per capita consumption of fruits and vegetables at the county level are related to the percentage of obese adults in the county. Data for this study are contained in the data file Food Nutrition Atlas, whose variable descriptions are found in the Chapter 9 appendix.

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