# MTH3003 SEMESTER PERTAMA 2012/13 ASSIGNMENT 1 Answer

MTH3003

SEMESTER PERTAMA 2012/13

ASSIGNMENT 1

Answer questions no 2,6,10, 13, 15, 16, 18, 20, 21, 22 and 24.

1. The ages (in months) at with 50 children were first enrolled in a preschool are listed below.

38 40 30 35 39 40 48 36 31 36

47 35 34 43 41 36 41 43 48 40

32 34 41 30 46 35 40 30 46 37

55 39 33 32 32 45 42 41 46 51

42 50 37 39 33 45 38 46 46 31 a.

Construct a stem and leaf display for the data. b.

Construct a relative frequency histogram for these data. Start the lower boundary of the first class at 30 and use a class width of 5 months. c.

Compare the graphs in parts a and b. Are there any significant differences that would cause you to choose one of the better method for displaying the data? d.

What proportion of the children were 35 months (2 years, 11 months) or older, but less than 45 months (3 years, 9 months) of age when first enrolled in preschool?

2. The number of calories per serving for selected ready-to-eat cereals is listed below.

130 190 140 80 100 120 220 220 110 100

210 130 100 90 210 120 200 120 10 120

190 210 120 200 130 180 260 270 100 160

190 240 80 120 90 190 200 210 190 180

115 210 110 225 190 130 a.

Construct a frequency distribution using 7 classes. b.

Draw a relative frequencies histogram for this data. Use the smallest value in the data set as lower class limit. c.

Find the mean and modal class. d.

Describe the shape of histogram.

3. You are given n = 8 measurements; 3, 2, 5, 6, 4, 4, 3, 5. a.

Find the mean, median, and mode. b.

Based on result in of mean and median, are the measurements symmetric or skewed? Draw a dotplot to confirm your answer.

4. You are given n = 8 measurements; 3, 1, 5, 6, 4, 4, 3, 5. a.

Calculate the range and sample mean. b.

Calculate the sample variance and standard deviation. c.

Compare the range and the standard deviation. The range is approximately how many standard deviations?

1

5. Given the following data set: 8, 7, 1, 4, 6, 6, 4, 5, 7, 6, 3, 0 a.

Find the five-number summary and the IQR. b.

Calculate 𝑥̅ and 𝑠 c.

Calculate the 𝑧 -score for the smallest and largest observations. Is either of these observations unusually large or unusually small?

6. The number of raisins in each of 14 mini boxes (1/2-ounce size) was counted for a generic brand and for Sunmaid brand raisins. The two of data sets are shown here:

25

Generic Brand

26 24

26

26

26

28

27

26

28

27

25

25

28

25

28

Sunmaid

29

24

28

24

24

28

30

24

25 28 22 27 a.

What are the mean and standard deviation for the generic brand? b.

What are the mean and standard deviation for the Sunmaid brand? c.

Compare the centers and variabilities of the two brands using the results of part a and b. d.

Find the median, the upper and lower quartiles, and the IQR for each of the two data sets. e.

Construct two box plots on the same horizontal scale to compare the two sets of data. f.

Draw two stem and leaf plots to depict the shapes of the two data sets. Do the box plots in part e verify the results?

7.

Two city council members are to be selected from a total of five to form a subcommittee to study the city’s traffic problems.

(a) How many different subcommittees are possible?

(b) If all possible council members have an equal chance of being selected, what is the probability that members Smith and Jones are both selected?

8.

A television news director wishes to use three news stories on an evening show. One story will be the lead story, one will be the second story, and the last will be a closing story. If the director has a total of eight stories to choose from, how many possible ways can the program be set up?

9.

An experiment consists of tossing a single die and observing the number of dots that show on the upper face. Event A, B, and C are defined as follows:

A : Observe a number less than 4

B : Observe a number less than or equal to 2

C : Observe a number greater than 3

Find the probabilities associated with the events below using either the simple event approach or the rules and definitions from this section.

(a) S (e) A

B

2

(b) A|B

(c) B

(d) A

B

C

(f) A

(g) A

(h) B

C

C

C

10. A smoker detector system uses two devices, A and B. If smoke is present, the probability that it will be detected by device A is 0.95; by device B is 0.98 and by both devices is 0.94.

(a) If smoke is present, find the probability that the smoke will be detected by device A or device B or both devices.

(b) Find the probability that the smoke will not be detected.

11.

The probability that Sam parks in a no-parking zone and gets a parking ticket is 0.06, and the probability that Sam cannot find a legal parking space and has to park in the no-parking zone is 0.20.

On Tuesday, Sam arrives at school and has to park in a no-parking zone. Find the probability that he will get a parking ticket.

12.

The probability that a patient is allergic to penicillin is 0.20. Suppose this drug is administered to three patients.

(a) Find the probability that all three of them are allergic to it.

(b) Find the probability that at least one of them is not allergic to it.

13.

Randomly selected 2000 adults were asked whether or not they have ever shopped on the Internet.

The following table gives a two-way classification of the responses obtained.

Male

Female

Have Shopped

300

200

Have Never Shopped

900

600

(a) If one adult is selected at random from these 2000 adults, find the probability that this adult i.

ii.

has never shopped on the Internet is a male iii.

has shopped on the Internet given that this adult is a female iv.

is a male given that this adult has never shopped on the Internet.

(b) Are the events “male” and “female” mutually exclusive? What the events “have shopped” and “male”? Why or why not?

(c) Are the events “female” and “have shopped” independent? Why or why not?

(d) Draw a tree diagram and find the following probabilities : i.

P (has never shopped on the Internet and is a male) ii.

P (has shopped on the Internet and is a female)

(e) Suppose that one adult is selected at random from these 2000 adults. Find the following probabilities. i.

P (has never shopped on the Internet or is a male)

3

ii.

P (is a male or has shopped on the Internet) iii.

P (has never shopped on the Internet or has never shopped on the internet)

14.

The following table lists the probability distribution of the number of customers visiting a convenience store per 15 minutes. x 5

0.12

6

0.20

8

0.18

9

0.15

10

0.08 p(x) a. Find p (7)

7

? b. Draw the probability histogram for this probability distribution. c. Find the probability that during a given 15-minute period, the number of customer visiting this store will be: i. Exactly 8 iii. Less than 8 ii. More than 7 iv. 7 to 9

15.

The following table gives the frequency distribution of the number of suits owned by all 100 managers of all companies in a city.

3 4 5 6 7 8 Number of suits

Number of managers

5 11 31 28 16 9 a.

Construct the probability distribution table for the number of suits owned by managers of these 100 companies. b.

Let x denotes the number of suits owned by a manager selected at random from these companies. Find the following probabilities. i. P( x = 5) iii. P( x

5) ii. P( x

6) iv. P(4

 x

7) v. P( x < 6) vi. P( x > 5) c. Find the mean and standard deviation for the probability distribution constructed in a.

16.

Car color preferences change over the years and according to the particular model that the customer selects. In a recent year, suppose that 10% of all luxury cars sold were black. If 25 cars of that year and type are randomly selected, find the following probabilities: a) At least 5 cars are black b) At most 6 cars are black c) More than 4 cars are black

4

d) Exactly 4 cars are black e) Between 3 and 5 cars (inclusive) are black f) More than 20 cars are not black

17.

Forty percent of all families own two or more cars. Let x be the number of families in a random sample of 15 families who own two or more cars. Find the mean and standard deviation of the probability distribution of x .

18.

On average, 6.2 customers visit a store per hour. Using Poisson probability distribution formula, find the probability that during a given hour exactly nine customers will visit the store?

19.

A washing machine in a Laundromat breaks down an average three times per month. What is the probability that during the next month this machine will have: a) Exactly two breakdowns b) At most one breakdown

20.

We have 100 items of which 12 are defectives. What is the probability that in a sample of 10, three are defectives?

21.

Dawn Corporation has 12 employees who hold managerial position. Of them, seven are female and five are male. The company is planning to send 3 of these 12 managers to a conference. If 3 managers are randomly selected out of 12, find the probability that; a) All 3 of them are female. b) At most 1 of them is a female.

22.

Suppose a random sample of n = 25 observations are selected from a population that is normally distributed with mean equal to 106 and standard deviation is equal to 12. a) Give the mean and standard deviation of the sampling distribution of the sample mean . b) Find the probability that exceeds 110. c) Find the probability that the sample mean deviates from the population mean μ = 106 by no more than 4.

23.

Mr. Ali, is an auditor in a credit card company, knows that in average a client have the balance in the account is RM 112 and standard deviation is RM 56. If he audited 50 accounts which he selected by random, what is the probability that the average balance in a month is below RM

100?

24.

Suppose that the population is known to be normally distributed, but the population mean is unknown. Suppose, however, that the population variance is known to be σ

2

= 16. What sample size is required for the sample mean to be within one unit of the true mean with 95% probability?

5