Statistics Papers

Allama Iqbal Open University, Islamabad Level: B.C.S Paper: Statistics for Management (471) Time Allowed: 3 Hours

Semester: Spring 2000 Maximum Marks: 100 Pass Marks: 33

Note: Attempt any FIVE Question. All carry equal marks. Q.1 Following the distance covered (in Km) by 45 staff office. 10 3 5 6 3 7 9 2 19 1 12 12 8 5 17 3 5 13 8 18 7 21 4 4

members of an office from their residence to the 8 12 11

6 10 16

11 9 19

20 2 3

15 1 6

13 11 17

13 10 16

(i) Construct a frequency distribution, by clearly mentioning all steps involved. (ii) Calculate mean and the Standard Deviation of the constructed frequency distribution. Q.2 a)

Assume that A and B are mutually exclusive events. Given that P(A) = 0.30 and P(B) = 0.40

6,6 3,3,4

∩

(i) What is P (A B)? (ii) What is P (A | B)? (iii) Do you agree with the argument that the concept of mutually exclusive events and independent events is same and that if events are mutually exclusive they must be independent? Use probability information in this problem to justify your answer. b)

In Nawabshah 60% of the Licensed drivers are 30 years of age or older and 40% of the drivers are under 30 years of age. Of all drivers under 30 years of age or older 4% will have traffic violation in a 12-month period. Assume that a driver under 30 years of age, 10% will have a traffic violation in 12-month period. Assume that a driver has just been charged with a traffic violation, what is the probability that the driver is under 30 years of age?

10

Q.3 a) A trial is repeated three times. Let X be the number of runs in the sequence of outcomes: first trail, 8 second trail and third trail. Find the probability distribution of X and graph it. b)

For the probability distribution constructed in the above part, calculate the following:

6

(i) E (X) (ii) Var (X) c)

A multiple choice test that has 30 questions and each one has five possible answers of which one is correct. If all the answers are guess, find the probability of getting exactly four correct answers.

Q.4 a) From the past experience the management of a well-known fast-food restaurant estimates that a number of weekly customers at a particular location is normally distributed with a mean of 5000 and a standard deviation of 800 customers. (i) What is the probability that on a given week the number of customers will be 4760 to 5800? (ii) What is the probability of more than 6500 customers?

6

5,5

b) In a courtroom in which a defendant is on trial for committing a crime with the assumption in the judicial system, a defendant is innocent until proven guilty.

3,4,3

(i) Formulate appropriate null and alternative hypothesis for judging the guilt or innocence of the defendant. (ii) Interpret the type-I and type-II errors in this context. (iii) If you are the defendant, would you want

α το βε σµαλλ ορ λαργε? Εξπλαιν.

Q.5 a) Suppose a manufacturer claims that his radial tires had a tread life of at least 40,000 k.m. A sample 8 of 49 tires was taken with mean 38000 k.m. Suppose it is known that the population of tire mileage has standard deviation of 3500 k.m. Test the hypothesis at 0.05 level of significance. b)

Q.6 a)

A consumer group wants to determine whether the mean time between failures of two products is the same. A sample of 36 failures of each product was examined. Product A had mean time between failures of 9 months and a variance of 3 months and a variance of 3 months. Product B had a mean time between failures of 12 months and variance of 4 months. At 0.02 level of significance test for a difference between the two products means. A study showed that 84 of 200 persons who saw shampoo advertisement during the telecast of cricket match and 96 of 200 persons who saw this advertisement during the Drama Hour remembered the name of the shampoo. Test the hypothesis at 0.05 level of significance that there is no difference between the corresponding true proportions.

b)

12

8

12 The question posed to 150 individuals that whether they would need a specified product. Their response are shown in the figure in the following contingency table. Set a relevant hypothesis of independence and test it. SEX Males

Females

Need

50

50

No Need

30

20

Total

80

70

Q.7 a) A research physician conducted an experiment to investigate the effects of various cold water temperatures on the pulse rate of small children. The data for seven 6 year children are: Temperature (X): Decrease in Pulse rate (Y):

6,6

68

65

70

62

60

55

58

2

5

1

10

9

13

10

(i) Find the least square line for the data. 2

(ii) Calculate S b) Find the coefficient of correlation between demand and supply of the following data: Supply: Demand:

400

200

700

100

500

300

600

50

60

20

70

40

30

10

Allama Iqbal Open University, Islamabad Level: B.A. Paper: Statistics for Management (471) Time Allowed: 3 Hours

Semester: Autumn 2000 Maximum Marks: 100 Pass Marks: 33

Note: Attempt any five questions. All carry equal marks. Q.1 a. b. Q.2

What information can be obtained from a frequency distribution.? Distinguish between percentile and quartile. The weights of a number of packages of frozen peas are given as follows:

10 10 20

1. 2. 3. 4. 5. Q.3 a.

b.

16.1 15.9 16.0 16.1 16.1 16.0 Make a frequency table. Draw a histogram. Find the mean. Find the median. Find the standard deviation.

15.8 16.0 16.0

16.3 16.1 15.9

16.2 16.0

16.0 15.9

Of six cars, produced at a particular factory between 8 an 10 A.M. last Monday morning, there are 20 known to be "lemons". Three of six cars were shipped to dealer A and the other three to dealer B. Just by chance, dealer A received all three lemons. What is the probability of this event occurring if, in fact, the three cars shipped to dealer A were selected at random from the six produced? Three coins are tossed and then following events are defined: A. {observe at least one head} B {observe exactly two heads} C. {observe exactly two tails} Assume that the coins are balanced. Calculate the following probabilities by summing the probabilities of the appropriate simple events.

∪ B) 3. P (A ∩ C)

∩ B) 4. P (B ∩ C)

1. P (P

2. P (A

5. Are events B and CC mutually exclusive? Q.4 a.

Suppose you invest a fixed sum of money in each of five business ventures. Assume you know that 20 70% of such ventures are successful, the outcomes of the ventures are independent of one another, and the probability distribution for the number, x, of successful ventures out of five is: X P(x)

0

1

2

3

4

5

.002

.029

.132

.309

.360

.168

µ = E (x) 2 b. Find σ = √ E [(x - µ) ] a. Find

b.

Q.5 a. b. Q.6

A machine that produces stampings for automobile engines is malfunctioning and producing 10% defectives. The defective and non-defective stampings proceed from the machine in a random manner. If the next five stampings are tested, find the probability that the three of them are defective. The expected number of minor injuries during a football match is 4.4. Find the probability that 20 during the course of a game there will be at the most three minor injuries. If the point estimator is biased but consistent, does this imply that the bias must approach zero as the sample size gets large? Explain. A manufacturer of cereal wants to test the performance of one of its filling machines. The machine 20

µ

is designed to discharge a mean amount of = 12 grams per box, and the manufacturer wants to detect any departure from this sitting. This quality control study calls for sampling 100 boxes to determine whether the machine is performing to specifications. Set up a test of hypothesis for this study, using Q.7 a.

b.

∝ = 0.01

The sales manager for a large appliance retailer, is measuring his radio advertising campaign 20 featuring major appliances. Over the last 7 weeks he has purchased varying amounts of radio time (line X, in minutes). Line Y displays the number of major appliances sold that week. X (minutes)

25

18

32

21

35

28

30

Y

16

11

20

15

26

32

20

1. Find the best-fitting line. 2. Calculate the standard error of the estimate Explain the difference between regression and correlation problems. Can a correlation also be a regression problem? Can regression problem also be a regression problem?

Allama Iqbal Open University Islamabad Paper: Statistics for Management (471) Level: B.A Time Allowed: 3 Hours.

Semester: Autumn 1999 Maximum Marks: 100 Pass Marks: 33

Note: ATTEMPT ANY FIVE QUESTIONS. ALL QUESTIONS CARRY EQUAL MARKS. Q.No.1 a). b).

Define the term Statistics. How a decision maker can benefit from the knowledge of Statistics. Following is the distribution of age of 100 persons given in the table. Age in Years:

15-29

30-39

40-44

45-49

50-54

55-59

Frequency

12

26

21

18

10

13

i). Draw the Histogram. ii). Draw Ogive. Q.No.2 a).

Let A = the event that a person holds a bachelor's degree B = the event that a person is a government employee C = the event that a person is an industrial worker. Further suppose that P(A) = 0.01, P(B) = 0.25 and P(C) = 0.20 1. 2.

b).

3. Find the probability that a person is not an industrial worker. Whirlwind Fans purchased capacitors from these suppliers A, B and C. Supplier A supplies 60% of the capacitors. B 30% and C 10%. The quality of the capacitors is known to vary among supplies. With A, B and C having 0.25%, 1% and 2% defective rates respectively. The capacitors are used in the fans produced by the company. 1.

Q.No.3 a).

What is the probability that a fan assembled by the company has a defective capacitor?

2. When a defective capacitor is found, which supplier is likely source? At a certain university it is known that 60 percent of the students are under 21 years of age. Also 50 percent of the students are female, and 30 percent of all students are both female and under 21. If one student is selected at random, what is the probability that the this student is: 1.

b).

Are A and B mutually exclusive? Can you find P(AB)? Are events B and C mutually exclusive. Find the probability that a person is a government employee or an industrial worker.

Female and under 21?

2. Male and 21 or over? A business firm is considering two investments of which it will choose one that promises the greater payoff. Which of the investment should it accept? (Let the mean profit measure the payoff.) Invest in Tool Shop

Q.No.4

a). b).

Q.No.5

a).

Q.No.6

a). b). c).

Invest in Tool Shop

Profit

Probability

Profit

Probability

Rs. 100,000

0.10

Rs. 400,000

0.20

50,000

0.30

90,000

0.10

20,000

0.30

-20,000

0.40

-80,000

0.30

-250,000

0.30

Total

1.00

Total

1.00

Data shows that in the age interval of 21 to 26, 40% of them cast their vote. If 16 individuals from that age group are randomly selected, find the probability that fewer than 2 of them vote. A traffic study conducted at one point on highway shows that vehicles speed (in Km/h) are normally distributed with a mean of 61.3 and a standard deviation of 3.3. If the vehicle is randomly checked, what is the probability that its speed is between 55 and 60 Km/h.? Distinguish between the following terms by giving examples: 1. Estimate and estimator. 2. Type I and type II arrors. 3. Null hypothesis and alternative hypothesis. Discuss the difference Point and Interval Estimation. Draw all possible sample of size 2 from a population of size 4 which is 0,2,3,5. Define the following, giving examples. 1.

Unbiased ness.

2.

Q.No.7

a).

b).

Confidence Interval.

3. Level of Significance The daily yield of a chemical recorded by a chemical manufacturing plant for 50 days, produce a mean and standard deviation of X = 871 and s = 21 tons. Test the hypothesis that the average daily yield of the chemical is µ = 880 tons per day against the alternative that µ is either greater or less than 880 tons/day. A pharmaceutical company that manufactures aspirin claims that 9 out of 10 doctors recommended aspirin for their patients. This test claim at a = 0.05 against the alternative that the actual proportion of doctors who do so is less than 90%, if a random sample of 100 doctors results in 80 who indicate that they recommended aspirin.

ALLAMA IQBAL OPEN UNIVERSITY, ISLAMABAD Paper: Statistics for Management (471) Level: B.A Time Allowed: 3 Hours

Semester: Autumn 1998 Maximum Marks: 100 Pass Marks: 33

Note: ATTEMPT FIVE QUESTIONS, AT LEAST TWO QUESTIONS FROM EACH PART. PART I Q.No.1

a)

Given the following information, indicate whether the distribution is symmetrical or skewed, also mention its direction. Mean = 40

Median = 50

4

Mode = 60

b)

Can the standard deviation ever be zero? Can it ever be negative? Explain.

4

c)

A supplier constructs a frequency table for the number of vat stereo units sold daily. Use that table to find the:

4+4+4

i

Mean

ii. Standard deviation

iii. Q1 and Q2 graphically No. Sold 0-3 4-7 8-11 12-15 16-19 Q.No.2

a)

Suppose A and B are mutually exclusive events and that P(A) = 0.45, P(b) = .35. i. Is the event A complement of the event B; Explain. ii. Find P(A and B); Explain. iii. Find P(A and B)c; Explain.

b)

In Nawab Shah 60% of the licensed drivers are 30 years of age or older and 40% of the 10 drivers are under 30 years of age. of all drivers 30 years of age or older 4% will have a traffic violation in a twelve month period. of all drivers under 30 years of age, 10% will have a traffic violation in a 12-month period. Assume that a driver has just been charged with a traffic violation; what is the probability that the driver is under 30 years of age? Suppose A and B are mutually exclusive events, P(A) = 0.60 and P(B) = 0.30. Are 4 events A andB independent. A retail store has received a shipment of six small appliances, two of which are 8,4 defective. A customer randomly selects and purchases two of the appliances as gifts. Find the probability distribution for the number of defective appliances among the two purchased. Also calculate the mean of the distribution. Explain clearly the difference between a binomial experiment and an hypergeometric 8 experiment, by giving an example? The fuses in a box of 10 fuses are defective. A sample of four fuses is examined without 3,3 replacement. Calculate the following probabilities.

c) Q.No.3

a)

b) Q.No.4

Frequency 5 9 8 6 3

a)

2,2,2

i. b) c)

Number defective fuse.

Specify K, so that i. P (Y > K) = 0.10

ii.

one or less defective fuse

ii.

P (74 - K < Y < 74 + K) = 0.90

3,3

A traffic study conducted at one point on highway shows that vehicles speed (in Km/h) 8 are normally distributed with a mean of 61.3 and a standard deviation of 3.3. If a vehicle is randomly checked, what is the probability that its speed is between 55 and 60 KM/h. PART II

Q.No.5

a)

Distinguish between the following terms by giving examples. i. Estimate and estimator ii. Type I and Type II errors. iii.

b)

Q.No.6

a)

Null hypothesis and alternative hypothesis.

A sample of 100 employees from a company was selected and the annual salary for 8 each was recorded. The mean and standard deviation were found to be 4000 (rupees) and 1000 (rupees) respectively. Construct a 95% confidence interval for the population average salary. using £ = 0.05, determine whether the mean tensile strengths in two shipments of 8 nylon cord are equal, given these sample observations:

Shipment I: Shipment II: b)

a)

b)

30 32

28 33

28 30

32 29

Before:

2.7

2.0

4.3

1.3

0.7

0.3 1.0 0.3 1.7

After:

1.7

1.3

1.7

1.0

0.3

0.3 0.3 0.0 0.0

12

At £ = 0.05 can you conclude that there was an improvement.

ii). Develop a 99% confidence interval for the change in mean number of cavities in population. In a study conducted at a large airport 81 of 300 persons who has just gotten off a plane and 32 of 200 persons who were about to board the plane admitted that they were afraid of flying. At £ = 0.05 test whether the difference between the sample proportions is significant. The following data were recorded about the dependence of hypertension smoking habits: Non Smokers Hypertension: No Hypertension:

Q.No.8

27 31

A toothpaste selling company is promoting a new product because of a recent study that showed a reduction in cavities after people used the toothpaste for six months period. A group of 10 people was selected and number of cavities were recorded before using the new toothpaste and after using it six months. The data are given below:

i).

Q.No.7

4,4,4

21 48

Moderate Smokers 36 26

8

12

Heavy Smokers 30 19

Test the hypothesis that the presence or absence of hypertension is independent of smoking habits. A research physician conducted an experiment to investigate the effects of various cold 5,5,5,5 water temperatures on the pulse rate of small children. The data are given as:

Temp. of water (XF): Decrease in pulse rate (Y beats/min): i. ii. iii. iv.

68

65

70

62

60 55 58

2

5

1

10

9 13 10

Find the least squares line for the data. Estimate the common errors variance. Construct a 95% confidence interval for B. Test the hypothesis Ho : B = 0 against H1 : B > 0.