Part A: Theoretical Questions (25 points) 1. ToysRUs wants to construct a confidence interval for the life of its toys. Which of the following confidence intervals will be wider: one based on a Z-distribution or one based on a t-distribution? 2. A confidence interval of 90% was used to estimate the proportion of customers who buy at least two items during their shopping mall experience. A random sample of 150 customers produced the following confidence interval: 30% /- 4%. What is the best way to explain the results? 3. When the population standard deviation is unknown and the sample size is less than 30 what table value should be used in computing a confidence interval for a mean? a. z b. t c. Chi-square d. None of the above 4. When you should use a z test? What are the properties of a t-distribution? 5. True or false a) Rejecting the null hypothesis when it is true is called a(n) Type I error. b) An error is committed when the null hypothesis is rejected when it is false. c) When you are conducting the t test the population must be approximately normally distributed. 6. For each conjecture state the null and alternative hypotheses: a) The average age of our customers is 38.2 years. b) The average age of the Chocolate-Fudge store customers is less than 14 years. 7. For each of the following claims mention what kind of test should be used (one-tail two-tail) and set up the null and alternate hypothesis. a) The Pastry shop claims that its chocolate cookies have at least 60% chocolate per cookie. b) The 87Cents Store claims that their daily profit is at least $1000. c) The Feather baggage store claims that its Blue_Sky brand luggage weights exactly 1.5 pounds. d) Coffee and Tea shop claims that their Latté has no more than 5 grams of fat. 8. What is the range of the values for the correlation coefficient? 9. Give examples of 2 variables that are positively correlated and two that are negatively correlated. 10. The mall has a set of data with employee age (X) and the corresponding number of annual on-thejob-accidents. Analysis on the set finds that the regression equation is Y=115-3.78X. What can be said of the correspondence between age and accidents? Are older workers safer or more prone to accident? What is the likely number of accidents for someone aged 25? What does the slope tell you about accidents relative to age? Part B: Calculation Problems (75 points) A mall is a big institution that must work efficiently especially during these slow-economic times. It contains many small shops and large departments. In order to make the mall a more efficient institution and a better shopping experience a large number of statistical tests have been conducted by the mall administration as well as by its shops. Most of the following problems refer to those statistical tests. 1. Human Resources: A recent study showed that working person experiences an average of 2.1 hours per day of distractions (phone calls e-mails text-messaging). A random sample of 50 employees for a large mall found that these employees were distracted an average of 1.9 hours per day and the population standard deviation was 20 minutes. Construct a 90% CIE of µ and compare your answers to the result of the study. 2. The Chocolate_Fudge shop claims that each of its cookies contains exactly 150 calories. You sample 100 cookies and find out that X = 155 calories and s = 10 calories. Should the store claim be rejected? Test the store’s claim at a = 0.1 3. The drug/vitamins store D&V claims that people using its sleeping pill will sleep for at least 8 hours a night. N=144 X = 7.5 hours s = 1.8 hours (a) Use the following sample to test at a = .05 (b) Suppose no claim was made but the company is thinking of making a claim using a 95% two-sided confidence interval. Construct the interval using the same data and suggest the maximum claim the company can make. 4. The drug/vitamins store D&V also claims that those using the Slimy drug for 30 days will lose at least 15 pounds. You sample 30 people who have used the drug and find that the average weight loss was 12 pounds with a standard deviation of 5 pounds. (a) Test the claim at the .05 significance level. (b) Assume that no claim was made by the company. They took a sample with the above results and ask you to construct a two-sided 95% confidence interval for the population mean. 5. Baterry&Gadgets store claims that its batteries have a life of 600 hours. The Mall Inspectors test a sample of 20 batteries and found: X = 580 hours s = 40 hours (a) Test at a=.05 (b) Construct a 2-tailed 95% C.I.E of µ 6. Human Resources took a survey and found that the average commute time one way is 25.4 minutes. However one of the executives feels that the commute is less. He randomly selects 25 commuters and finds that the average is 22.1 minutes with a standard deviation of 5.3 minutes. At a=0.10 is he correct? 7. ToysRUs and the Teddy Bear stores claimed that their toys last longer. Which store toys last longer? Test at .05 significance level ToysRUs: Average toy life= 14.50 years; standard deviation = 1.50 years; n = 10 TeddyBear store claimed: Average toy life = 13.60 years; standard deviation = 2.10 years; n = 11 8. Another text done by the human resources was about how long different categories of employees are working (hours /pay period). Who works longer married or unmarried employees? Test at a =.01 Unmarried Employee Married Employee 140 160 14 0. 16 0. 78 5. 77 0. 1 2 1 2 1 2 = = = = ? = ? = n n S hours S hours hours hours 9. The data below shows the number of beverages bought during a week by the employees from their cafeteria (using the employee discount card) and their hourly salary. Calculate the correlation coefficient and explain the obtained result. Hourly Wage (X) Beverages (Y) $12 50 $14 30 $15 20 $13 20 $15 18 $14 13 $20 10 $19 4 $22 0 $25 0 ?X = 169; ?Y = 165 ?XY = 2308; ?X 2 = 3025; ?Y 2 = 4809 10. A women’s clothing department wants to find out if there is a relationship (and what kind of relationship) between the price of an item and the time that it takes to a customer to decide in buying that item. Item Price Decision Time (in minutes) $30 15 $20 18 $300 30 $80 20 $70 40 $25 30 $17 10 $63 20 $150 32 $250 40 $10 5 $8 20 $42 10 $187 25 Calculate the following statistically useful quantities. ?X ?Y ?XY ?X 2 ?Y 2 Calculate the correlation coefficients. R R 2 Is there a strong correlation? Create a scatter plot using MS Excel and discuss the obtained results. 11. The Tea&Coffee shop is advertising their green tea supply. They claim that the more green tea you drink the longer you leave. We want to see if the quantity of drinking green tea (per week) really affects people longevity. Write out the regression equation. What is the correlation coefficient? What is r-squared (coefficient of determination)? Discuss the results. Green Tea Quantity (X) Longevity (Y) 0 70 0 68 0 75 2 66 2 76 3 72 4 69 4 73 6 72 6 74 8 72 8 77 10 73 10 77 12 76 17 78 20 81 24 82 30 86 32 89 ?X = 198; ?Y = 1506; ?XY = 15 890; ?X 2 = 3 782; ? Y2 = 114 048
