Work standards specify time, cost, and efficiency norms for the performance of work tasks. They are typically used to monitor job performance. In one distribution center, data were collected to develop work standards for the time to assemble or fill customer orders. The table below contains data for a random sample of 9 orders.

1. Construct a simple linear regression of the data using SPSS. What is the equation of the least squares line? Use time as the dependent variable.
2. In general, we would expect the mean time to fill an order to increase with the size of the order. Do the data support this theory? Test using α = .05.

5. You’re competing in the rock paper scissors international tournament. Use the following information to determine whether your semifinal opponent over-represents certain choices (?=0.05). Show all relevant values behind your decision.

6. 	Frequency
   Rock	85
   Paper	110
   Scissors	105
   Total	300

A population of values has a normal distribution with μ=62.7μ=62.7 and σ=66.2σ=66.2. You intend to draw a random sample of size n=42n=42.

Find the probability that a single randomly selected value is greater than 49.4.
P(X > 49.4) =

Find the probability that a sample of size n=42n=42 is randomly selected with a mean greater than 49.4.
P(M > 49.4) =

Enter your answers as numbers accurate to 4 decimal places. Answers obtained using exact z-scores or z-scores rounded to 3 decimal places are accepted.

License

Consider the experiment of rolling a six-sided fair die. Let X

denote the number of rolls it takes to obtain the first 5,

Y denote the number of rolls until the first 2, and Z denote

the number of rolls until the first 4. Numerical answers are needed only for parts (a) and

(b). Expressions are sufficient for parts (c), (d), and (e).

a) E[X|Y = 1 or Z = 1]

b) E[X|Y = 1 and Z = 2]

c) E[X|Y = 1 and Z = 3]

d) E[X|Y = 3 and Z = 4]

e) E[X^2 |Y = 3 and Z = 4]

Question 1 [35 marks]

The purchasing manager has formulated the following LP model:

Minimise COST = 6M1 + 5M2 + 4M3 + 7M4

subject to

0,05M1 + 0,04M2 + 0,08M4 ≥ 50 (CHRM)

0,40M1 + 0,80M2 + 0,60M3 + 0,32M4 ≤ 500 (IRON)

M1 + M2 + M3 + M4 = 1000 (MASS)

and all variables ≥ 0,

where Mi = number of kg of scrap type i purchased, i=1,2,3,4.

(a) Solve this model using LINDO or SOLVER.

(b) Write down the foundry's optimal purchasing plan and cost.

Based on your LINDO or SOLVER solution answer the following questions by using only the initial printout of the optimal solution. (This means that you may not change the relevant parameters in the model and do reruns.)

(c) How good a deal would the purchasing manager need to get on scrap type 1 before he would be willing to buy it for this order?

(d) Upon further investigation, the purchasing manager finds that scrap type 2 is now being sold at R5,40 per kg. Will the purchasing plan change? By how much will the cost of purchasing the metals increase?

(e) The customer is willing to raise the ceiling on the iron content in order to negotiate a reduction in the price he pays for the order. How should the purchasing manager react to this?

(f) The customer now specifies that the alloy must contain at least 6% chromium. Can the purchasing manager comply with this new specification? Will the price charged for the order change?

The mean of a population is 74 and the standard deviation is 16. The shape of the population is unknown. Determine the probability of each of the following occurring from this population.

a. A random sample of size 32 yielding a sample mean of 76 or more

b. A random sample of size 130 yielding a sample mean of between 72 and 76

c. A random sample of size 220 yielding a sample mean of less than 74.3

A population of values has a normal distribution with μ=81.3μ=81.3 and σ=88.7σ=88.7. You intend to draw a random sample of size n=168n=168.

Find P₈₀, which is the score separating the bottom 80% scores from the top 20% scores.
P₈₀ (for single values) =

Find P₈₀, which is the mean separating the bottom 80% means from the top 20% means.
P₈₀ (for sample means) =

Enter your answers as numbers accurate to 1 decimal place.
************NOTE************ round your answer to ONE digit after the decimal point! ***********
Answers obtained using exact z-scores or z-scores rounded to 3 decimal places are accepted.

• 10.7 When people make estimates, they are influenced by anchors to their estimates. A study was conducted in which students were asked to estimate the number of calories in a cheeseburger. One group was asked to do this after thinking about a calorie-laden cheesecake. A second group was asked to do this after thinking about an organic fruit salad. The mean number of calories estimated in a cheeseburger was 780 for the group that thought about the cheesecake and 1,041 for the group that thought about the organic fruit salad. (Data extracted from “Drilling Down, Sizing Up a Cheeseburger's Caloric Heft,” The New York Times, October 4, 2010, p. B2.) Suppose that the study was based on a sample of 20 people who thought about the cheesecake first and 20 people who thought about the organic fruit salad first, and the standard deviation of the number of calories in the cheeseburger was 128 for the people who thought about the cheesecake first and 140 for the people who thought about the organic fruit salad first.

• a. State the null and alternative hypotheses if you want to determine whether the mean estimated number of calories in the cheeseburger is lower for the people who thought about the cheesecake first than for the people who thought about the organic fruit salad first.

• b. In the context of this study, what is the meaning of the Type I error?

• c. In the context of this study, what is the meaning of the Type II error?

• d. At the 0.01 level of significance, is there evidence that the mean estimated number of calories in the cheeseburger is lower for the people who thought about the cheesecake first than for the people who thought about the organic fruit salad first?

. A manager for an insurance company believes that customers have the following preferences for life insurance products: 40% prefer Whole Life, 10% prefer Universal Life, and 50% prefer Life Annuities. The results of a survey of 310 customers were tabulated. Is it possible to refute the sales manager's claimed proportions of customers who prefer each product using the data?

Product Number Whole 124 Universal 31 Annuities 155

State the null and alternative hypothesis.

What does the null hypothesis indicate about the proportions of fatal accidents during each month?

State the null and alternative hypothesis in terms of the expected proportions for each category.

Find the value of the test statistic. Round your answer to three decimal places.

Find the degrees of freedom associated with the test statistic for this problem.

Find the critical value of the test at the 0.025 level of significance. Round your answer to three decimal places.

Make the decision to reject or fail to reject the null hypothesis at the 0.025 level of significance.

State the conclusion of the hypothesis test at the 0.025 level of significance.  

Write the null and alternative hypotheses in notation for each of the following statements.

Given the following information, test whether the population mean is equal to the given value μ₀. Provide the following:

17. The appropriate test statistic (z_obs or t_obs)
17. The critical value of the test statistic
17. A statement of your conclusion (i.e., is the null rejected?)

Use for Questions 1-7: Hector will roll two fair, six-sided dice at the same time. Let A = the event that at least one die lands with the number 3 facing up. Let B = the event that the sum of the two dice is less than 5.

1. What is the correct set notation for the event that “at least one die lands with 3 facing up and the sum of the two dice is less than 5”?

2. Calculate the probability that at least one die lands with 3 facing up and the sum of the two dice is less than 5.

3. What is the correct set notation for the event that “at least one die lands with 3 facing up if the sum of the two dice is less than 5”?

4. Calculate the probability that at least one die lands with 3 facing up if the sum of the two dice is less than 5.

5. What is the correct set notation for the event that “the sum of the two dice is not less than 5 if at least one die lands with 3 facing up”?

6. Calculate the probability that the sum of the two dice is not less than 5 if at least one die lands with 3 facing up.

7. Are A and B independent? Explain your reasoning

A sample of 50 night-school students' ages is obtained in order to estimate the mean age of night-school students. x = 24.3 years. The population variance is 16.

(b) Find the 95% confidence interval for μ. (Give your answer correct to two decimal places.)

Lower Limit

Upper Limit

(c) Find the 99% confidence interval for μ. (Give your answer correct to two decimal places.)

Lower Limit

Upper Limit

Solve the problem.

21) The total home-game attendance for major-league baseball is the sum of all attendees for all stadiums during the entire season. The home attendance (in millions) for a number of years is shown in the table below.

21)

Year

Home Attendance (millions)

1978

40.6

1979

43.5

1980

43.0

1981

26.6

1982

44.6

1983

46.3

1984

48.7

1985

49.0

1986

50.5

1987

51.8

1988

53.2

a) Make a scatterplot showing the trend in home attendance. Describe what you see.

b) Determine the correlation, and comment on its significance.

c) Find the equation of the line of regression. Interpret the slope of the equation.

d) Use your model to predict the home attendance for 1998. How much confidence do you have in this prediction? Explain.

e) Use the internet or other resource to find reasons for any outliers you observe in the

scatterplot.