# statistics homework help

Determine whether or not the matrix is stochastic.

Yes

No

2. –/9 pointsGoldFM10 8.1.010.

Write a stochastic matrix corresponding to the transition diagram.

A B C

A

B

C

HW 10 (Homework)

Peter Rubenstein MAT 183, section 100, Spring 2016 Instructor: Vincent Fatica

WebAssign

3. –/9 pointsGoldFM10 8.1.011.

Write a stochastic matrix corresponding to the transition diagram.

A B C

A

B

C

4. –/9 pointsGoldFM10 8.1.012.

Write a stochastic matrix corresponding to the transition diagram.

A B C

A

B

C

5. –/11 pointsGoldFM10 8.1.014.

A retailer stocks three brands of breakfast cereal. A survey is taken of 5000 people who purchase cereal weekly from this retailer. Each week Crispy Flakes loses 10% of its customers to Crunchy Nuggets and 15% to Toasty Cinnamon Twists. Crunchy Nuggets loses 16% of its customers to Crispy Flakes and 11% of its customers to Toasty Cinnamon Twists, and Toasty Cinnamon Twists loses 19% of its customers to Crispy Flakes and 14% to Crunchy Nuggets.

(a) Set up a stochastic matrix displaying these transitions.

F N T

F

N

T

A

(b) Suppose that this week 1500 people buy Crispy Flakes, 1000 buy Crunchy Nuggets, and 2500 people buy Toasty Cinnamon Twists. How many people will buy Crispy Flakes next week?

(c) How many people will buy Toasty Cinnamon Twists in two weeks?

6. –/2 pointsGoldFM10 8.1.019.

Referring to Example 5, consider a typical group of French women, of whom 46% currently work. Assume that the same percentage of daughters follow in their mothers’ footsteps as with the American women-that is, those given by the following matrix, A = matrix(2,2[.8,.3,.2,.7])

Use A and A2 to determine the proportion of working women in the next two generations. (Round off to the nearest whole percent.)

(a) After one generation, % of French women will work.

(b) After two generations, % of French women will work.

7. –/3 pointsGoldFM10 8.1.021.

Taxis pick up and deliver passengers in a city that is divided into three zones. Records kept by the drivers show that of the passengers picked up in zone I, 50% are taken to a destination in zone I, 40% to zone II, and 10% to zone III. Of the passengers picked up in zone II, 40% go to zone I, 30% to zone II, and 30% to zone III. Of the passengers picked up in zone III, 20% go to zone I, 60% to zone II, and 20% to zone III.

If originally 32% of the taxis start in zone I, 47% in zone II, and 21% in zone III, how will the taxis be distributed after each has taken one passenger? (Round your answer to 1 decimal place.)

% (zone I) % (zone II) % (zone III)

8. –/13 pointsGoldFM10 8.1.024.

For a certain group of states, it was observed that 80% of the Democratic governors were succeeded by Democrats and 20% by Republicans. Also, 20% of the Republican governors were succeeded by Democrats and 80% by Republicans.

(a) Set up the 2 × 2 stochastic matrix with columns and rows labeled D and R that displays these transitions.

D R

D

R

A

(b) Compute A2 and A3.

A2 =

A3 =

(c) Suppose that all the current governors are Democrats. Assuming that the current trend holds for three elections, what percent of the governors will then be Democrats?

9. –/12 pointsGoldFM10 8.1.025.

Each day mice are put into a T-maze (a maze shaped like a “T”; shown below). In this maze they have the choice of turning to the left (rewarded with cheese) or to the right (receive cheese along with mild shock). After the first day their decision whether to turn left or right is influenced by what happened on the previous day. Of those that go to the left on a certain day, 73% go to the left on the next day and 27% go to the right. Of those that go to the right on a certain day, 64% go to the left on the next day and 36% go to the right.

(a) Set up the 2 × 2 stochastic matrix with columns and rows labeled L and R that describes this situation.

L R

L

R

A

(b) Compute the second power of the matrix in part (a)

A2 =

(c) Suppose that on the first day (day 0) 50% go to the left and 50% go to the right. So, the initial distribution is given by the following column matrix.

matrix(2,1[.5,.5])

Using the matrices in parts (a) and (b), find the distribution matrices for the next two days, days 1 and 2.

x1 =

x2 =

(d) Make a guess as to the percentage of mice that will go to the left after 50 days. (Do not compute. Do this on paper. Your instructor may ask you to turn in this work.)

10.–/10 pointsGoldFM10 8.1.026.

A group of physical fitness devotees works out in the gym every day. The workouts vary from strenuous to moderate to light. When their exercise routine was recorded, the following observation was made: Of the people who work out strenuously on a particular day, 30% will work out strenuously on the next day and 70% will work out moderately. Of the people who work out moderately on a particular day, 50% will work out strenuously and 50% will work out lightly on the next day. Of the people working out lightly on a particular day, 20% will work out strenuously on the next day, 10% moderately, and 70% lightly.

(a) Set up the 3 × 3 stochastic matrix with columns and rows labeled S, M, and L that describes these transitions.

S M L

S

M

L

(b) Suppose that on a particular Monday 50% have a strenuous, 40% a moderate, and 10% a light workout. What percent will have a strenuous workout on Wednesday?

%

11.–/5 pointsGoldFM10 8.1.027.

According to the Higher Education Research Institute,† 32% of students at baccalaureate granting colleges who entered college in 2007 characterize their political views as Liberal, 43% as Middle-of-the- road, and 25% as Conservative. Suppose that each year these students changed their political views as described by the following matrix.

(a) What percentage of the students who held middle-of-the-road political views as freshmen held liberal views as sophomores?

%

(b) Explain the meaning of the percentage .96 appearing in the center of the matrix.

From view: L M C

To view: L M C

.94 .02 .01

.05 .96 .04

.01 .02 .95

96% of the freshman who held middle-of-the-road political views continued to hold these views as sophomores.

96% of the freshman who held middle-of-the-road political views changed these views by the time they were sophomores.

96% of the freshman who held middle-of-the-road political views continued to hold these views as seniors.

96% of the freshman who held middle-of-the-road political views continued to hold these views as juniors.

(c) Draw the transition diagram for this Markov process.

(d) What percentage of the students held liberal political views as sophomores? (Round your answer to two decimal places.)

%

What percentage of the students held liberal political views as juniors? (Round your answer to two decimal places.)

%

12.–/5 pointsGoldFM10 8.1.028.

According to the Higher Education Research Institute,† 80% of students at baccalaureate granting colleges who entered college in 2007 lived in College residence halls, 13% lived with Family, and 7% lived in Other types of housing. Suppose that each year these students changed their residences as described by the following matrix.

(a) What percentage of students who lived in college residence halls as freshmen lived with family as sophomores?

%

(b) Explain the meaning of the percentage .8 appearing in the center of the matrix. 80% of the freshman who lived with family continued to do so as sophomores.

80% of the freshman who lived with family continued to do so as seniors.

80% of the freshman who lived with family continued to do so as juniors.

80% of the freshman who lived with family had other living arrangements by the time they were sophomores.

(c) Draw the transition diagram for this Markov process.

From residence: C F O

To residence: C F O

.9 .1 .2

.05 .8 .1

.05 .1 .7

(d) What percentage of the students lived with family as sophomores? (Round your answer to one decimal place.)

%

What percentage of the students lived with family as juniors? (Round your answer to one decimal place.)

%

13.–/10 pointsGoldFM10 8.1.029.

A sociologist studying living patterns in a certain region determines that each year the population shifts between urban, suburban, and rural areas as shown in the figure below.

(a) Set up a stochastic matrix A that displays these transitions.

U S R

U

S

R

A

(b) What percentage of people who live in urban areas in 2006 will live in rural areas in 2008? %

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