A supermarket chain analyzed data on sales of a particular brand of snack cracker

at 104 stores for a certain one week period. The analyst decided to build a regresion model to predict the sales of the snack cracker based on the total sales of all brands in the snack cracker category.

f. Produce a 90% confidence prediction interval for the sales of the cracker

   in a store where the category sales is 1005. Also produce the90 %

   confidence prediction interval in a store where category sales is 900.

  Can you with 90% confidence claim which store has higher cracker sales?

Use the data given in the table to answer the following questions. The data represents the average number of miles that a salesperson travels in a day verses the number of sales made each month.

(a) What is the value of the correlation coefficient for this set of data? Round to 3 decimal places.

(b) What is the equation of the Regression Line for this set of data? Round values to two decimal places.

(c) Predict the number of sales an associate could expect to make if he travelled an average of 108 miles each day. Round to two decimal places.

Corporate advertising tries to enhance the image of the corporation. A study compared two ads from two sources, the Wall Street Journal and the National Enquirer. Subjects were asked to pretend that their company was considering a major investment in Performax, the fictitious sportswear firm in the ads. Each subject was asked to respond to the question "How trustworthy was the source in the sportswear company ad for Performax?" on a 7-point scale. Higher values indicated more trustworthiness. Here is a summary of the results. Ad source n x s Wall Street Journal 66 4.77 1.50 National Enquirer 61 2.43 1.64 Find the two-sample pooled t statistic. Then formulate the problem as an ANOVA and report the results of this analysis. Verify that F = t 2.

1.13. Problem. (Section 3.4) Three black boxes are labeled with Roman numerals I, II and III. • Box I contains four red chips and three blue chips. • Box II contains two red chips and five blue chips. • Box III contains seven red chips and no blue chips. Solve each of the following problems.

(a) Suppose a box is selected at random and three chips are drawn at random from the box. If all three chips are red, what is the probability they were drawn from Box I?

(b) Suppose one chip is selected at random from each box. If two of the three chips drawn are red chips, what is the probability that the chip drawn from Box II was red?

(c) Suppose three chips are randomly selected from Box I and placed in Box III. If a chip subsequently drawn randomly from Box III is blue, what is the probability that all three chips moved from Box I to Box III were blue.

Evaluate the differences between dependent and independent samples. Would our random samples need to come from the same "overall" population?

1.14. Problem. (Section 3.4) A public health researcher examines the medical records of a group of 937 men who died in 1999 and discovers that 210 of the men died from causes related to heart disease. Moreover, 312 of the 937 men had at least one parent who suffered from heart disease, and, of these 312 men, 102 died from causes related to heart disease. Determine the probability that a man randomly selected from this group died of causes related to heart disease, given that neither of his parents suffered from heart disease. Upon arrival at a hospital’s emergency room, patients are categorized according to their condition as critical, serious, or stable. In the past year,

• 10% of ER patients were critical

• 30% of ER patients were serious

• 60% of ER patients were stable

• 40% of critical patients died.

• 10% of the serious patient died.

• 1% of the stable patients died. Given a patient survived, what is the probability that they were categorized as serious upon arrival?

1.11. Problem. (Sections 2.2-2.4, 3.1) Five cards are drawn from a standard deck of 52 cards.

(a) Given that exactly three of the five cards show a hearts suit, calculate the probability that the hand also includes a three-of-a-kind.

(b) Given that the five card hand contains a three-of-kind, find the probability that it contains at three hearts.

Six people of different ages are getting in line to buy coffees. Compute the number of ways they can arrange themselves in line such that no three consecutive people are in increasing order of age, from front to back

1.8. Problem. (Sections 2.2-2.4) Three fair, six-sided dice colored red, green and blue are rolled. Calculate each of the following probabilities:

(a) The probability all three dice show the same face (“triples”).

(b) The probability that the red die shows a larger number than the green die.

(c) The probability that the red die shows a larger number than the green die and the green die shows a larger number than the blue die.

(d) The probability that the sum of the pips on all three dice is exactly 10.

(e) The probability that the sum of the pips on all three dice is less than 10.

(f) The probability that the sum of the pips on all three dice is greater than 10.

Discuss the three properties (characteristics) of data and explain some of the descriptive measures associated with each property.

1. The length of industrial filters is a quality characteristic of interest. Thirty samples,
   each of size 5, are chosen from the process. The data yields an average length of
   110 mm, with the process standard deviation estimated to be 4 mm.
   (a) Find the warning limits for a control chart for the average length.
   (b) Find the 3sigma control limits. What is the probability of a type I error?
   (c) If the process mean shifts to 112 mm, what are the chances of detecting this shift
   by the third sample drawn after the shift?
   (d) What is the chance of detecting the shift for the first time on the second sample
   point drawn after the shift?
   (e) What is the ARL for a shift in the process mean to 112 mm? How many samples,
   on average, would it take to detect a change in the process mean to 116 mm?

The landing of military fighter jets on aircraft carrier requires great skill, so on occasions it requires more than one attempt to achieve the landing. TOP GUN is a pilot who is assigned to an aircraft carrier and has a record of achieving 95 % of landings on aircraft carriers in the first attempt. In a particular exercise TOP GUN is assigned to make four (4) takeoffs and landings on the aircraft carrier to which it is assigned. Under the assumption that the resulting events in each landing attempt are statistically independent of each other determine:

Most import: without using the binomial distribution. *NO BINOMIAL DISTRIBUTION*

a) The probability that TOP GUN achieve four (4) landings in the first (1) try.

b) The probability that TOP GUN achieve at least one (1) landing out of the four (4) on the first try.

This is an extension of the Birthday Problem. Suppose you have 500 Facebook friends. Make the same assumptions here as in the Birthday Problem.

a) Write a program in R to estimate the probability that, on at least 1 day during the year, Facebooks tells you three (or more) of your friends shat that birthday. Based on your answer, should you be surprised by this occurrence?

b) Write a program in R to estimate the probability that, on at least 1 day during the year, Facebook tells you five (or more) of your friends share that birthday. Based on your answer, should you be surprised by the occurrence? [Hint: Generate 500 birthdays with replacement, then determine whether any birthday occurs three or more times (five or more for part (b)). The table function in R may prove useful.]

The mean waiting time at the drive-through of a fast-food restaurant from the time an order is placed to the time the order is received is 84.3 seconds. A manager devises a new drive-through system that he believes will decrease wait time. He initiates the new system at his restaurant and measures the wait time for ten randomly selected orders. The wait times are provided in the table below. Based on the given data, is the new system effective? Use the α = 0.10 level of significance.

On a separate sheet of paper, write down the hypotheses (H₀ and H_a) to be tested.

Conditions:
Use Minitab Express to perform a normality test on the given data.
The P-value for the Anderson-Darling test of normality is ______ (Report this value exactly as it appears in Minitab Express. Do not round.)
Based on both the normal probability plot and this P-value, the t-test for means  (is / is not) valid for the given data.

Rejection Region:
To test the given hypotheses, we will use a (left / right / two) -tailed test. The appropriate critical value(s) for this test is/are _________ .  (Report your answer exactly as it appears in Table VI. For two-tailed tests, report both critical values in the answer blank separated by only a single space.)

Hypothetical Human population matrix over a period of time of 20 year intervals.

Age 0-20 20-40 40-60 60-80   

0-20 [ .24 .98 0 0]

20-40 [ .77 0 .92 0]

40-60 [ .04 0 0 .57]

60-80 [ 0 0 0   0]

Complete the calculation to determine what the population distribution will be 200 years after the initial probability distribution shown in the example as P= [1000, 1000, 1000, 1000]. The formula is P*T^10 (there are 10 sets of 20 in 200)

After 200 years, # of people in the 0-20 range = _______________?

After 200 years, # of people in the 20-40 range = _______________?

After 200 years, # of people in the 40-60 range = _______________?

After 200 years, # of people in the 60-80 range = _______________?

Now, using the same population dynamics matrix, determine what the probability distribution will be after 320 yrars if the initial probability distribution is P= [1100, 1700, 1100, 1000] ? The formula is P*T^16 (there are 16 sets of 20 in 320)

After 320 years, # of people in the 0-20 range = _______________?

After 320 years, # of people in the 20-40 range = _______________?

After 320 years, # of people in the 40-60 range = _______________?

After 320 years, # of people in the 60-80 range = _______________?