A retailer discovers that 3 jars from his last shipment of Spiffy peanut butter contained between 15.85 and 15.92 oz of peanut butter, despite the labeling indicating that each jar should contain 16 oz. of peanut butter. He is wondering if Spiffy is cheating its customers by filling its jars with less product than advertised. He decides to measure the weight of 50 jars from the shipment and use hypothesis testing to verify this.

(a) What are the null and alternative hypotheses for this experiment?

(b) Describe, in words, a Type I error for this experiment.

(c) Describe, in words, a Type II error for this experiment.

(d) Given the answer to (a), should the null hypothesis be rejected when the sample mean falls below or over a certain threshold? Should this threshold be below or above the value 16.0 oz?

(e) What is the distribution of X¯, the sample mean?

(f) In his sample of 50 jars, the retailer finds an average weight of 15.84 oz and a sample standard deviation of 0.5 oz. He decides to use a significance level of 0.04. What is the conclusion from this hypothesis testing? Can you conclude that Spiffy is cheating its customers?

(g) What is the p-value? What is the meaning of this number?

(h) For what values of the sample mean would the null hypothesis be rejected?

(i) Calculate the probability of type II error if the true mean is 15.7 oz.

(j) Solve (f), (h) and (i) when the level of significance is 0.01. Is your new answer for (f) consistent with the p-value found in (g)? How is the probability of type II error affected when the probability of type I error changes?

A retailer discovers that 3 jars from his last shipment of Spiffy peanut butter contained between 15.85 and 15.92 oz of peanut butter, despite the labeling indicating that each jar should contain 16 oz. of peanut butter. He is wondering if Spiffy is cheating its customers by filling its jars with less product than advertised. He decides to measure the weight of 50 jars from the shipment and use hypothesis testing to verify this.

(a) What are the null and alternative hypotheses for this experiment?

(b) Describe, in words, a Type I error for this experiment.

(c) Describe, in words, a Type II error for this experiment.

(d) Given the answer to (a), should the null hypothesis be rejected when the sample mean falls below or over a certain threshold? Should this threshold be below or above the value 16.0 oz?

(e) What is the distribution of X¯, the sample mean?

(f) In his sample of 50 jars, the retailer finds an average weight of 15.84 oz and a sample standard deviation of 0.5 oz. He decides to use a significance level of 0.04. What is the conclusion from this hypothesis testing? Can you conclude that Spiffy is cheating its customers?

(g) What is the p-value? What is the meaning of this number?

(h) For what values of the sample mean would the null hypothesis be rejected?

(i) Calculate the probability of type II error if the true mean is 15.7 oz.

(j) Solve (f), (h) and (i) when the level of significance is 0.01. Is your new answer for (f) consistent with the p-value found in (g)? How is the probability of type II error affected when the probability of type I error changes?

A retailer discovers that 3 jars from his last shipment of Spiffy peanut butter contained between 15.85 and 15.92 oz of peanut butter, despite the labeling indicating that each jar should contain 16 oz. of peanut butter. He is wondering if Spiffy is cheating its customers by filling its jars with less product than advertised. He decides to measure the weight of 50 jars from the shipment and use hypothesis testing to verify this.

(a) What are the null and alternative hypotheses for this experiment?

(b) Describe, in words, a Type I error for this experiment.

(c) Describe, in words, a Type II error for this experiment.

(d) Given the answer to (a), should the null hypothesis be rejected when the sample mean falls below or over a certain threshold? Should this threshold be below or above the value 16.0 oz?

(e) What is the distribution of X ̄, the sample mean?

(f) In his sample of 50 jars, the retailer finds an average weight of 15.84 oz and a sample standard deviation of 0.5 oz. He decides to use a significance level of 0.04. What is the conclusion from this hypothesis testing? Can you conclude that Spiffy is cheating its customers?

(g) What is the p-value? What is the meaning of this number?

(h) For what values of the sample mean would the null hypothesis be rejected?

(i) Calculate the probability of type II error if the true mean is 15.7 oz.

(j) Solve (f), (h) and (i) when the level of significance is 0.01. Is your new answer for (f) consistent with the p-value found in (g)? How is the probability of type II error affected when the probability of type I error changes?

Problem 11-15
Risky Cash Flows

The Bartram-Pulley Company (BPC) must decide between two mutually exclusive investment projects. Each project costs $6,750 and has an expected life of 3 years. Annual net cash flows from each project begin 1 year after the initial investment is made and have the following probability distributions:

BPC has decided to evaluate the riskier project at a 12% rate and the less risky project at a 8% rate.

What is the expected value of the annual net cash flows from each project? Do not round intermediate calculations. Round your answers to nearest dollar.

What is the coefficient of variation (CV)? Do not round intermediate calculations. (Hint: ?B=$5,097 and CVB=$0.70.)

What is the risk-adjusted NPV of each project? Do not round intermediate calculations. Round your answer to the nearest dollar.

If it were known that Project B is negatively correlated with other cash flows of the firm whereas Project A is positively correlated, how would this affect the decision?
This would tend to reinforce the decision to -Select-acceptrejectItem 9 Project B.

If Project B's cash flows were negatively correlated with gross domestic product (GDP), would that influence your assessment of its risk?
-Select-YesNoItem 10

Risky Cash Flows

The Bartram-Pulley Company (BPC) must decide between two mutually exclusive investment projects. Each project costs $8,000 and has an expected life of 3 years. Annual net cash flows from each project begin 1 year after the initial investment is made and have the following probability distributions:

BPC has decided to evaluate the riskier project at a 12% rate and the less risky project at a 10% rate.

1. What is the expected value of the annual net cash flows from each project? Do not round intermediate calculations. Round your answers to nearest dollar.

   	Project A	Project B
   Net cash flow	$	$

   
   What is the coefficient of variation (CV)? Do not round intermediate calculations. (Hint: σ_B=$6,158 and CV_B=$0.78.)

   	σ (to the nearest whole number)	CV (to 2 decimal places)
   Project A	$
   Project B	$

2. 
   
3. What is the risk-adjusted NPV of each project? Do not round intermediate calculations. Round your answer to the nearest dollar.

   Project A		$
   Project B		$

4. 
   
5. If it were known that Project B is negatively correlated with other cash flows of the firm whereas Project A is positively correlated, how would this affect the decision?
   This would tend to reinforce the decision to

Project B.

If Project B's cash flows were negatively correlated with gross domestic product (GDP), would that influence your assessment of its risk?

Lyft has recently launched “Lyft Scooters” in various cities across the U.S. Imagine the company selects two individuals, Zahra in Madison, Wisconsin, and Mateo in Albuquerque, New Mexico, to test Lyft Scooters in their respective cities. Each recognizes that she or he has a 2% chance of experiencing an accident. If an accident occurs, a $12,000 will be lost due to injury and property damage.

1. What is Zahra’s probability distribution for loss arising out of an accident?

2. What is Zahra’s expected value of loss from an accident?

3. What is the standard deviation of Zahra’s probability distribution of loss arising out of an accident?

For questions 4-11, assume that Zahra and Mateo decide to pool (or share equally) their losses. The losses are uncorrelated.

4. How much will each person need to pay into the pot of pooled money in order to cover their losses in the event they both experience an accident?

5. What is the combined probability distribution when the two loss potentials are pooled?

6. What is the expected total loss for the distribution of pooled losses?

7. What is the expected loss per person from the distribution of pooled losses?

8. What is the standard deviation of the distribution of pooled losses?

9. What is the standard deviation of the distribution of pooled losses, per person?

10. If the number of people in the pool increases from 2 to 1000, how will this change influence each of the following? Use words, not numbers, to state whether they will they increase, decrease or stay the same.
    
    1. The expected total loss
       
       
    2. The expected loss per person
       
       
    3. The standard deviation of the total loss
       
       
    4. The standard deviation of loss per person

Based on your work so far, why might Zahra and Mateo choose to pool their losses?

1. Let  z be a normal random variable with mean 0 and standard deviation 1. What is P(-2.25 <  z < -1.1)?

You are offered an investment opportunity. Its outcomes and probabilities are presented in the following table.

2. The mean of this distribution is _____________.

3. T/F. The probability that the complement of an event will occur is given by P(E') = 1 - P(E)

True

False

4.

A recent survey of local cell phone retailers showed that of all cell phones sold five years ago, 64% had a camera, 28% had a music player, and 22% had both. The probability that a cell phone sold five years ago did not have either a camera or a music player is

5.

The sample standard deviation is related to the sample variance through what functional form?

6.

A large industrial firm allows a discount on any invoice that is paid within 30 days. Of all invoices, 10% receive the discount. In a company audit, 10 invoices are sampled at random. The binomial probability that fewer than 3 of the 10 sampled invoices receive the discount is approximately_______________.

7.

Suppose x is a normal random variable with mean 60 and standard deviation 2. A z score was calculated for a number, and the z score is 3.4. What is the inverse normal calculation of x?

Risky Cash Flows

The Bartram-Pulley Company (BPC) must decide between two mutually exclusive investment projects. Each project costs $8,000 and has an expected life of 3 years. Annual net cash flows from each project begin 1 year after the initial investment is made and have the following probability distributions:

BPC has decided to evaluate the riskier project at a 11% rate and the less risky project at a 9% rate.

1. What is the expected value of the annual net cash flows from each project? Do not round intermediate calculations. Round your answers to nearest dollar.

   	Project A	Project B
   Net cash flow	$	$

   
   What is the coefficient of variation (CV)? Do not round intermediate calculations. (Hint: σ_B=$6,158 and CV_B=$0.78.)

   	σ (to the nearest whole number)	CV (to 2 decimal places)
   Project A	$
   Project B	$

2. 
   
3. What is the risk-adjusted NPV of each project? Do not round intermediate calculations. Round your answer to the nearest dollar.

   Project A		$
   Project B		$

4. 
   
5. If it were known that Project B is negatively correlated with other cash flows of the firm whereas Project A is positively correlated, how would this affect the decision?
   This would tend to reinforce the decision to -Select-acceptreject Project B.
   
   If Project B's cash flows were negatively correlated with gross domestic product (GDP), would that influence your assessment of its risk?
   -Select-YesNo

Problem 3-31 (Algorithmic)

Gulf Coast Electronics is ready to award contracts to suppliers for providing reservoir capacitors for use in its electronic devices. For the past several years, Gulf Coast Electronics has relied on two suppliers for its reservoir capacitors: Able Controls and Lyshenko Industries. A new firm, Boston Components, has inquired into the possibility of providing a portion of the reservoir capacitors needed by Gulf Coast. The quality of products provided by Lyshenko Industries has been extremely high; in fact, only 0.5% of capacitors provided by Lyshenko had to be discarded because of quality problems. Able Controls has also had a high quality level historically, producing an average of only 2% unacceptable capacitors. Because Gulf Coast Electronics has had no experience with Boston Components, it estimated Boston Components’ defective rate to be 9%. Gulf Coast would like to determine how many reservoir capacitors should be ordered from each firm to obtain 73000 acceptable-quality capacitors to use in its electronic devices. To ensure that Boston Components will receive some of the contract, management specified that the volume of reservoir capacitors awarded to Boston Components must be at least 12% of the volume given to Able Controls. In addition, the total volume assigned to Boston Components, Able Controls, and Lyshenko Industries should not exceed 29000, 48000, and 47500 capacitors, respectively. Because of Gulf Coast’s long-term relationship with Lyshenko Industries, management also specified that at least 29500 reports should be ordered from Lyshenko. The cost per capacitor is $2.4 for Boston Components, $2.5 for Able Controls, and $2.8 for Lyshenko Industries.

1. Formulate and solve a linear program for determining how many reservoir capacitors should be ordered from each supplier to minimize the total cost of obtaining 75,000 acceptable-quality reservoir capacitors. For subtractive or negative numbers use a minus sign even if there is a + sign before the blank. Enter "0" if your answer is zero. If the constant is "1" it must be entered in the box.
   
   Let B = number of capacitors ordered from Boston Components
   
   Let A = number of capacitors ordered from Able Controls
   
   Let L = number of capacitors ordered from Lyshenko Industries
   

   Min	B	+	A	+	L
   s.t.
   	B					≤		Boston
   			A			≤		Able
   					L	≤		Lyshenko
   	B ____	+	A____	+	L	=		# useful capacitors
   	B	+	A			≥		Boston - Able %
   					L	≥		Minimum Lyshenko
   B, A, L ≥ 0

   
   Round your answers to the nearest whole number.
   

   Optimal Solution:
   B	____
   A	____
   L	____

2. Suppose that the quality level for reservoir capacitors supplied by Boston Components is much better than estimated. What effect, if any, would this quality level have?
   
   If the quality of reservoir capacitors supplied by Boston Components is much better than expected the optimal solution may change. For example, if Boston Components reduced their defect rate from 9% to 2%, the new optimal solution would increase the number of reservoir capacitors given to Boston Components to .
   
3. Suppose that management is willing to reconsider their requirement that at least 30,000 capacitors must be ordered from Lyshenko Industries. What effect, if any, would this consideration have? If required, round your answer to the nearest cent.
   
   If management removes or reduces the requirement that Lyshenko Industries be ordered at least 30,000 reservoir capacitors the cost will decrease by $__________ per unit decreased.