A queueing system serves two types of customers. Type 1 customers arrive according to a Poisson process with a mean rate of 5 per hour. Type 2 customers arrive according to a Poisson process at a mean rate of 3 per hour. The system has two servers, both of which serve both types of customer. All service times have exponential distribution with a mean of 10 minutes. Service is provided on a first-come-first-served basis.

a. What is the probability distribution of the time between consecutive arrivals of customers of any type, what is its mean?

b. Assume that when a Type 2 customer arrives, he finds two Type 1 customers being served and no other customers in the system. What is the probability distribution of this Type 2 customer’s waiting time in the queue and its mean?

A high street store was interested to discover if the spending of customers using credit cards is different from the spending of customers using cash. A random sample of 52 customers using credit cards for a single transaction had a mean spend of £36.43 with standard deviation £12.08. A random sample of 38 customers using cash for a single transaction had a mean spend of £31.84 with standard deviation £11.27.

A hypothesis test is to be performed to investigate whether the mean spend by customers using credit cards was equal to the mean spend by customers using cash.

(a) Using appropriate notation, which you should define, specify the null and alternative hypotheses associated with the test.

(b) Calculate the value of the estimated standard error of the difference between the sample means, as well as the value of the test statistic.

(c) Complete the hypothesis test, carefully detailing the conclusions of the test.

The manager of the Burrard Credit Union wishes to know if there is a significant difference
between male and female customers’ interest in a proposed new type of savings certificate. A
survey of 200 randomly selected customers has yielded the following data:

INTEREST

SEX Strong Moderate Weak Total
Male 30 25 25 80
Female 60 40 20 120
Total 90 65 45 200

(a) What percentage of customers shows a strong or moderate interest in these savings
certificates? [2 marks]

ANSWER

(b) What percentage of customers is male AND shows a strong or moderate interest in
these savings certificates? [2 marks]

ANSWER

(c) What percentage of male customers do NOT show a strong interest in these savings
certificates? [2 marks]

ANSWER

(d) Among those customers who show a weak interest in these types of savings certificates,
what is the probability that they are a female?

A queueing system serves two types of customers. Type 1 customers arrive according to a Poisson process with a mean rate of 5 per hour. Type 2 customers arrive according to a Poisson process at a mean rate of 3 per hour. The system has two servers, both of which serve both types of customer. All service times have exponential distribution with a mean of 10 minutes. Service is provided on a first-come-first-served basis.

a. What is the probability distribution of the time between consecutive arrivals of customers of any type, what is its mean?

b. Assume that when a Type 2 customer arrives, he finds two Type 1 customers being served and no other customers in the system. What is the probability distribution of this Type 2 customer’s waiting time in the queue and its mean?

A monopolist faces 300 customers divided into 3 different groups:

1. High-Demand customers each have a demand function given by

Q_H = 18 - P.

2. Medium-Demand customers each have a demand function given by

Q_M = 16 - P.

3. Low-Demand customers each have a demand function given by

Q_L = 14 - P.

There are 100 customers of each type (N_H=N_M=N_L=100).

The marginal cost of producing (one unit of) the product the firm is selling is constant at MC = $4. There is no fixed cost.

1. Determine the optimal two-part tariff for this firm and the resulting profits (it can only select one two-part tariff that is applied to all customers).

2. Do the same for N_H=N_L=50 and N_M=200. Explain briefly the difference between the results in (1) and (2).

The NFL Scouting Combine is an event at which football scouts evaluate the abilities of some top college prospects. Following are heights in inches and weighs in pounds for some of the quarterbacks at the 2013 Combine. Compute the slope b 1 least-square regression line for predicting weight from height. Round your answer to 3 decimal places.

Height (x) 75 78 75 79 75 76 77 77 74 75 74 76 75 74

weight (y) 227 232 231 225 240 225 226 237 227 219 218 237 226 215

=

The NFL Scouting Combine is an event at which football scouts evaluate the abilities of some top college prospects. Following are heights in inches and weighs in pounds for some of the quarterbacks at the 2013 Combine. Compute the slope LaTeX: b1 least-square regression line for predicting weight from height. Round your answer to 3 decimal places.

Height (x) 75 78 75 79 75 76 77 77 74 75 74 76 75 74

weight (y) 227 232 231 225 240 225 226 237 227 219 218 237 226 215

A simple random sample of checks were categorized based on the number of cents on the written check and recorded below. Cents Category 0¢-24¢ 25¢-49¢ 50¢-74¢ 75¢-99¢ Frequency 58 37 28 17 Use the p-value method and a 5% significance level to test the claim that 50% of the check population falls into the 0¢-24¢ category, 20% of the check population falls into the 25¢-49¢ category, 16% of the check population falls into the 50¢-74¢ category, and 14% of the check population falls into the 75¢-99¢ category. Calculate the expected value for the 50¢-74¢ category (round to the nearest tenth).

The data below shows height​ (in inches) and pulse rates​ (in beats per​ minute) of a random sample of women. Construct a​ scatterplot, find the value of the linear correlation coefficient​ r, and find the​ P-value using alphaequals0.05. Is there sufficient evidence to conclude that there is a linear correlation between height and pulse​ rate?

What are the null and alternative​ hypotheses?

A.

Upper H 0H0​:

rhoρequals=0

Upper H 1H1​:

rhoρgreater than>0

B.

Upper H 0H0​:

rhoρequals=0

Upper H 1H1​:

rhoρless than<0

C.

Upper H 0H0​:

rhoρequals=0

Upper H 1H1​:

rhoρnot equals≠0

D.

Upper H 0H0​:

rhoρnot equals≠0

Upper H 1H1​:

rhoρequals=0

Construct a scatterplot. Choose the correct graph below.

A.

64807290xy

A scatterplot has a horizontal x-scale from less than 64 to 80 in intervals of 2 and a vertical y-scale from less than 72 to 90 in intervals of 2. Twenty points are plotted with coordinates as follows: (60.5, 71); (61.5, 71); (63.5, 68); (58, 73); (60, 79); (60.5, 82); (59.5, 70); (59.5, 83); (63, 63); (60.5, 70); (61, 86); (60.5, 60); (64, 66); (64.5, 79); (66, 78); (67, 71); (68, 84); (67.5, 66); (69.5, 78); (65, 75). All horizontal coordinates are approximate.

B.

64807290xy

A scatterplot has a horizontal x-scale from less than 64 to 80 in intervals of 2 and a vertical y-scale from less than 72 to 90 in intervals of 2. Twenty points are plotted with coordinates as follows: (54, 65); (55, 68); (57, 66); (58, 69); (58, 71); (59, 70); (61, 72); (63, 76); (64, 74); (66, 78); (67, 77); (68, 75); (70, 79); (71, 80); (71, 82); (73, 80); (73, 84); (74, 83); (76, 88); (76, 85).

C.

64807290xy

A scatterplot has a horizontal x-scale from less than 64 to 80 in intervals of 2 and a vertical y-scale from less than 72 to 90 in intervals of 2. Twenty points are plotted with coordinates as follows: (52, 76); (52, 84); (52, 87); (53, 62); (53, 82); (53, 87); (56, 63); (56, 85); (58, 77); (63, 62); (64, 72); (65, 62); (65, 88); (69, 64); (69, 82); (70, 78); (74, 68); (76, 76); (77, 63); (77, 78).

D.

64807290xy

A scatterplot has a horizontal x-scale from less than 64 to 80 in intervals of 2 and a vertical y-scale from less than 72 to 90 in intervals of 2. Twenty points are plotted with coordinates as follows: (52, 86); (53, 84); (54, 79); (55, 82); (56, 82); (57, 82); (58, 78); (60, 80); (61, 77); (62, 76); (64, 78); (65, 72); (66, 75); (68, 70); (69, 73); (71, 68); (72, 70); (74, 67); (77, 62); (77, 65).

The linear correlation coefficient r is

nothing.

​(Round to three decimal places as​ needed.)

The test statistic t is

nothing.

​(Round to three decimal places as​ needed.)

The​ P-value is

nothing.

​(Round to three decimal places as​ needed.)

Is there sufficient evidence to conclude that there is a linear correlation between the two​ variables?

A.

YesYes​,

because the​ P-value is

lessless

than the significance level.

B.

YesYes​,

because the​ P-value is

greatergreater

than the significance level.

C.

NoNo​,

because the​ P-value is

greatergreater

than the significance level.

D.

NoNo​,

because the​ P-value is

lessless

than the significance level.

The following data relates to the size of the electricity bill sent to 7 randomly selected customers and the
time the customers took to pay the bills.
Customer 1 2 3 4 5 6 7
Size of bill($) 1500 1800 2300 2700 3300 3700 4600
Time to pay (days) 16 21 19 20 24 30 27
a) State the dependent variable and independent variable of the data set.
b) Plot the data on a scatter graph and add a trend line on the scatter graph.
c) Find the correlation coefficient and comment on it.
d) Determine the linear regression equation that can be used to predict how long a bill of a given size
will take to pay.
e) Interpret your equation.
f) Use your equation to predict how long it will take a customer to pay:
i. A bill of $1050, and
ii. A bill of $8000.
What reservations do you have about these predictions?