CMTH 380, TEST 1, Summer 2024

CMTH 380, TEST 1, Summer 2024

Duration: 3 hours + 30 minutes: from 6 pm to 9.30 pm

Last Name ___________________________________First Name: ___________________________ Student ID Number _________________________

Total mark:

50

1. (4 marks) The data set of 18 values is given below 3.2, 4.0, 4.2, 2.1, 4.4, 3.8, 4.2, 3.3, 2.8, 3.9, 2.8, 3.5, 2.6, 3.7 2.2, 4.3, 3.2, 2.7 Construct stem and leaf plot for this data set by using each number in the ones place twice.

Indicate the leaf unit.

Leaf unit = _______________ 1

2. (4 marks) The data set of ages (in years) of 25 employees of a company is given below.

27, 28, 28, 28, 29, 30, 32, 33, 34, 35, 37, 38, 39, 40, 41, 42, 43, 45, 45, 47, 48, 49, 50, 52, 53 Construct frequency and relative frequency distribution for this data set with the class width 5.

Class interval

Frequency

Relative frequency

The total of all frequencies is ____________________ The total of all relative frequencies is ____________________ 3. ( 6 marks) The length of time it takes for a worker to perform certain task (in minutes) was recorded for 10 workers. The resulting data values were the following: 4, 11 , 3, 1, 7, 2, 4, 6, 8, 10 Find the mean, median, mode, range, variance and standard deviation. In the answers for the mean, variance, and standard deviation leave one digit after decimal point. a. Mean = ____5.8____

b. Median = ____5.5_____

d. Range = ____10____

e. Variance = __9.6__

c. Mode = ___4______

f. Standard deviation = __3.1____

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4. (4 marks) Fill in missing values in the following statements.

The distribution of scores on the test written by a large group of students is skewed with the mean of 72 points and standard deviation of 5 points.

Fill in missing values in the following statements. a. According to Tchebysheff’s Theorem, at least 89% of scores will fall into the interval between ______57_____ points and _______87_______ points. b. At least _____75_____ percent of scores fall between 62 points and 82 points. c. At most _____25_____ percent of scores are smaller than 62 points or larger than 82 points. d. At most _____12.5_____ percent of scores are smaller than 62 points.

5. (3 marks) Fill in missing values in the following statements.

The distribution of the lengths of long-stemmed roses is mound – shaped and symmetric with the mean stem length of 15 inches and standard deviation of 2 inches.

According Empirical rule a. ____68___ percent of the long-stemmed roses have the lengths between 13 inches and 17 inches. b. ____16___ percent of long-stemmed roses have the lengths smaller than 13 inches. c. ____81.5___ percent of long-stemmed roses have the lengths between 11 inches and 17 inches.

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6. (3 marks) Write missing values in the statements a. and b. Write proper names in the statement c.

Alex and Boris are taking college preparatory exams.

Alex writes the ACT test and scores 20 out of 35. Boris write SAT test, and scores 1050 out of 1600.

If the means and standard deviations of scores for the ACT test are 25 and 5, and for the SAT test are 1150 and 50, then a. Alex’s z -score is ________________; b. Boris z -score is ________________; c. ______________ performed on exam better than ______________

7. (2 marks) Write missing values in the statements a. and b. a. If Alex has scored 510 on the MCAT, and was at the 80th percentile, and Boris has scored 520, and was at the 90th percentile, then approximately ___________% of all the students who wrote MCAT had the score lower than Alex or higher than Boris. b. If 20% of U.S. women have the height larger than 5. 8 feet, and 60% of U.S. women have the

weight less than 170 pounds, then 170 is ________ th percentile and 5.8 is ________ th percentile.

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8. (4 marks) The following data set contains the exam marks of 20 students in a calculus class.

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58 69 71 72 72 74 75 76 78 78 80 81 81 82 82 88 89 94 98

a. Find the median, the first and the third quartiles, and the interquartile range. b. Find the left outer fence, left inner fence, right inner fence, and right outer fence. c. Find the length of left whisker and the length of right whisker in the box – whisker plot. d. List all outliers (if any) in the last row. If there are no outliers state “none”.

List all extreme values (if any) in the last row. If there are no extreme values state “none”.

Place calculated value in the table given below. a.

Q = 1

Median =

b.

LOF =

LIF =

c.

LLW =

LRW =

d.

Outliers:

Extreme values:

Q = 3

RIF =

IQR = ROF =

5

9. (4 marks) Three people are randomly selected to report for jury duty. Each selected person is as likely to be a male as a female. Assume that that males (m) and females (f) are equally likely. a. The sample space of random experiment is

List outcomes of the following events and find their probabilities. b. The event “three females are selected” is B =_________________ The probability of event B is ____________________ c. The event “at most one female is selected” is C =____________________________ The probability of event C is ____________________ d. The event “at least one female is selected” is D =______________________________________________ The probability of event D is ____________________

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10. (1 + 2 = 3 marks) An unbalanced six – side die is rolled once and the number on the top of the die is observed.

Suppose that probability to get number 2 is twice as large as to get number 1, and probability to get each of numbers 3, 4, 5, and 6 is 0.1.

Let A be the event “the number obtained on the top of the die is odd (1 or 3 or 5)” Let B be the event “the number obtained on the top of the die is smaller than 4(1 or 2 or 3)” a. The probability of event A is ________. b. The probability of event B is ________.

11. (3 marks) Suppose that you close your eyes and select at random two candies from the bowl which contains 2 red and 3 yellow candies without replacement. Use counting rules to find the following probabilities. a. Probability that you will select one red and one yellow candy is____________ b. Probability that you select two yellow candies is ____________

12. (3 marks) Four people, A, B, C and D, are equally qualified runners. They run a 500 – meter sprint, and the order of finish is recorded,

a. Probability that A comes the last is___________. b. Probability that A wins and D comes the third is ___________. c. Probability that runners will come in order A, B, C, and D or in order D, C, B, A is ____________

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13. (4 marks) A pair of balanced dice is rolled once and the numbers on the top of the dice are observed. a. Probability that the sum of numbers on the tops of the dice is 10 is_____________ b. Probability that the numbers on the top of the dice are the same is _____________ c. Probability that the 2nd number in the pair is larger than the 1st number is _____________ d. Probability that the pair of numbers on the tops of the dice contains at least one digit 2 is _________________

14. (3 marks) a. You play card game and select two cards from 52 - card deck with replacement (you replace the card after the first selection and shuffle the deck).

Probability that two cards selected by you are different is ___________ b. You play card game and select two cards from 52 - card deck with replacement (you replace the card after the first selection and shuffle the deck).

Probability that two cards selected by you are the same is ___________ c. You play card game and select two cards from 52 - card deck without replacement.

Probability that two cards selected by you are “spades” is ___________ (there are 13 spades in the deck)

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