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For each of Questions 1 to 5, is the given value of the correlation coefficient reasonable?

Hint: think about the strength and the direction of the relationship between the two variables in each case.

Note: It is subjective to decide whether the magnitude of the correlation between two variables should be, for example, 0.7 or 0.8. The below questions don't ask you to make a decision like this.

Question 1 

A correlation of r = +0.7 between gender and height.

Question 2 

Let X = speed of vehicles driving on the highway and Y = distance for the vehicles to stop when the brakes are applied.

A correlation of r = +1.0 between X and Y.

Question 3 

A correlation of r = 0 between number of slurpees (a frozen beverage) sold at 7-Eleven in one day and number of cups of hot chocolate sold at the same store in the same day.

Question 4

A correlation of r = +0.7 between number of pages in a fiction novel and time it takes to read the novel.

Question 5 

Let X = time it takes a marathon runner to finish the race and Y = number of runners who finish ahead of him.

A correlation of r = -0.8 between X and Y.

1. Given  P(A) = 0.3 and P(B) = 0.2, do the following. (For each answer, enter a number.)

(a) If A and B are mutually exclusive events, compute P(A or B).

(b) If P(A and B) = 0.3, compute P(A or B).

2. Given P(A) = 0.8 and P(B) = 0.4, do the following. (For each answer, enter a number.)

(a) If A and B are independent events, compute P(A and B).

(b) If P(A | B) = 0.1, compute P(A and B).

3.  The following question involves a standard deck of 52 playing cards. In such a deck of cards there are four suits of 13 cards each. The four suits are: hearts, diamonds, clubs, and spades. The 26 cards included in hearts and diamonds are red. The 26 cards included in clubs and spades are black. The 13 cards in each suit are: 2, 3, 4, 5, 6, 7, 8, 9, 10, Jack, Queen, King, and Ace. This means there are four Aces, four Kings, four Queens, four 10s, etc., down to four 2s in each deck.

You draw two cards from a standard deck of 52 cards without replacing the first one before drawing the second.

(a) Are the outcomes on the two cards independent? Why?

No. The probability of drawing a specific second card depends on the identity of the first card.

Yes. The probability of drawing a specific second card is the same regardless of the identity of the first drawn card.   

No. The events cannot occur together.

Yes. The events can occur together.

(b)Find P(ace on 1st card and nine on 2nd). (Enter your answer as a fraction.)

(c) Find P(nine on 1st card and ace on 2nd). (Enter your answer as a fraction.)

(d) Find the probability of drawing an ace and a nine in either order. (Enter your answer as a fraction.)

4.  

You draw two cards from a standard deck of 52 cards, but before you draw the second card, you put the first one back and reshuffle the deck.

(a)Are the outcomes on the two cards independent? Why?

Yes. The probability of drawing a specific second card is the same regardless of the identity of the first drawn card.

Yes. The events can occur together.    

No. The events cannot occur together.

No. The probability of drawing a specific second card depends on the identity of the first card.

(b) Find P(ace on 1st card and king on 2nd). (Enter your answer as a fraction.)

(c)  Find P (king on 1st card and ace on 2nd). (Enter your answer as a fraction.)

(d)Find the probability of drawing an ace and a king in either order. (Enter your answer as a fraction.)

7. Consider the following scenario:

• Let P(C) = 0.2

• Let P(D) = 0.3

• Let P(C | D) = 0.4

• Part (a)
  Find P(C AND D).

• Part (b)
  Are C and D mutually exclusive? Why or why not?C and D are not mutually exclusive because
  P(C) + P(D) ≠ 1
  .C and D are mutually exclusive because they have different probabilities. C and D are not mutually exclusive because
  P(C AND D) ≠ 0
  .There is not enough information to determine if C and D are mutually exclusive.

• Part (c)
  Are C and D independent events? Why or why not?The events are not independent because the sum of the events is less than 1.The events are not independent because
  P(C) × P(D) ≠ P(C | D)
  . The events are not independent because
  P(C | D) ≠ P(C)
  .The events are independent because they are mutually exclusive.

• Part (d)
  Find P(D | C).

8. G and H are mutually exclusive events.

• P(G) = 0.5

• P(H) = 0.3

• Part (a)
  Explain why the following statement MUST be false:
  P(H | G) = 0.4.
  The events are mutually exclusive, which means they can be added together, and the sum is not 0.4.The statement is false because P(H | G) =

• = 0.6. To find conditional probability, divide
  P(G AND H) by P(H)
  , which gives 0.5.The events are mutually exclusive, which makes
  P(H AND G) = 0
  ; therefore,
  P(H | G) = 0.

• Part (b)
  Find
  P(H OR G).

• Part (c)
  Are G and H independent or dependent events? Explain

• G and H are dependent events because they are mutually exclusive.

• G and H are dependent events because

• P(G OR H) ≠ 1.

• G and H are independent events because they are mutually exclusive.

• There is not enough information to determine if G and H are independent or dependent events.

9.

Approximately 281,000,000 people over age five live in the United States. Of these people, 55,000,000 speak a language other than English at home. Of those who speak another language at home, 62.3 percent speak Spanish.

• E = speaks English at home

• E' = speaks another language at home

• S = speaks Spanish at home

Finish each probability statement by matching the correct answer.

• Part (a)
  P(E' )
  = ---Select--- 0.1219 0.1957 0.6230 0.8043

• Part (b)
  P(E)
  = ---Select--- 0.1219 0.1957 0.6230 0.8043

• Part (c)
  P(S and E' )
  = ---Select--- 0.1219 0.1957 0.6230 0.8043

• Part (d)
  P(S | E' )
  = ---Select--- 0.1219 0.1957 0.6230 0.8043

Consider historical data showing that the average annual rate of return on the S&P 500 portfolio over the past 85 years has averaged roughly 8% more than the Treasury bill return and that the S&P 500 standard deviation has been about 29% per year. Assume these values are representative of investors' expectations for future performance and that the current T-bill rate is 4%.

Calculate the utility levels of each portfolio for an investor with A = 3. Assume the utility function is U = E(r) − 0.5 × Aσ². (Negative amounts should be indicated by a minus sign. Do not round intermediate calculations. Round your answers to 4 decimal places.)

Consider historical data showing that the average annual rate of return on the S&P 500 portfolio over the past 85 years has averaged roughly 8% more than the Treasury bill return and that the S&P 500 standard deviation has been about 37% per year. Assume these values are representative of investors' expectations for future performance and that the current T-bill rate is 5%.

Calculate the utility levels of each portfolio for an investor with A = 2. Assume the utility function is U = E(r) − 0.5 × Aσ². (Do not round intermediate calculations. Round your answers to 4 decimal places.)

Consider historical data showing that the average annual rate of return on the S&P 500 portfolio over the past 85 years has averaged roughly 8% more than the Treasury bill return and that the S&P 500 standard deviation has been about 24% per year. Assume these values are representative of investors' expectations for future performance and that the current T-bill rate is 3%.

Calculate the utility levels of each portfolio for an investor with A = 2. Assume the utility function is U = E(r) − 0.5 × Aσ². (Do not round intermediate calculations. Round your answers to 4 decimal places.)

Table 1

Calculate the following for the car park using the data from Table 1:

a) the maximum real power demand P (kW)

b) maximum reactive power demand Q (kVAr)

c) the total apparent power demand S (kVA)

d) also, calculate the maximum current magnitude |I| if the supply voltage magnitude |V| is 3.3 kV and the supply frequency is 50 Hz.

e) Calculate the overall car park power factor.

f) Calculate the rating of the capacitor bank (μF) needed to improve the car park power factor to unity.

g) Calculate the new maximum current magnitude, after power factor correction. Again assume the supply voltage magnitude is 3.3kV and frequency is 50 Hz.

h) Select the cable ID from Table 2 that you would use to supply the car park BEFORE AND AFTER power factor correction has been applied.

i) Comment on your result.

Table 2

1. Five samples of table legs produced in an automated cutting process were taken each hour for 20 hours. The length of a table leg in centimeters was measured, and yielded averages and ranges show in the below table. Assume the sample size (n) is 5.

1. 1. Use the data to construct an chart and an R chart. Show all steps and all your work, including: Overall Mean and Overall Average Range; Calculations for Center Line, Upper Control Limit, and Lower Contol Limit, Visual representation of the control chart (either plotted in Excel or drawn to scale on graph paper). You must show all work for full credit.
   2. Is the process in control or out of control? Explain why thoroughly.
2. A fast-food franchise tracked the number of errors that occurred in customers’ orders. These included wrong menu item, wrong drink size, lack of condiments, wrong price total, etc. Some orders may have had more than one error. In one week, 1250 orders were filled, and a total of 30 errors were discovered. Find the Midline (, and the Upper and Lower Control Limits for a c chart. Show all your work for full credit.