# HCC Highway Engineer Worksheet

Math 3339Review for Test 1 KEY

1.

2.

3.

4.

P(X ≥ 5) = 0.8338

5.

6.

7.

8.

0.33

a. 0.1

b. 0.1

c. yes

a. 0.027

b. 0.778

9.

a.

b. 0.82

c. 0.16

10. 0.5

11.

12.

13.

a.

b.

c.

d.

e.

P(X = 4) = 0.15

2.25

1.178

3

a.

b.

c. 4.2

d. 17.64

a. 192

b. 108

c. 0.4375 or 7/16

14.

a.

b. No

15.

a. 0.044

b. 0.3409

16.

0.9066543

17. 0.111

18.

a)

b) Strength: Moderate, Direction: positive, Form: linear

c) Correlation = r = 0.561 there is a positive moderate relationship between the grades

on the first exam (x) and the grades on the final exam (y).

d) LSLR: 𝑦𝑦� = 12.0623 + 0.7771𝑥𝑥

e) X = 85; predicted final exam score = 78.1158

f) Coefficient of determination: R2 = 0.3147, this means that 31.47% of the variation

in the final exam scores can be explained by the LSLR. This low of a R2 implies

that the first exam score may not be the best (or only thing) to predict final exam

score.

19. This is binomial with n = 15 and p = 0.05

a) P(X = 5) = 0.00056

b) This is a low probability.

c) E(X) = 15*.05 = 0.75 (which also confirms that having 5 defective may be too

many defective).

20. Population size 17, 9 men and 8 women, sample size 5.

a) Hypergeometric

b) P(3 men and 2 women) = 0.3801

21. X = cost of man’s dinner, Y = cost of woman’s dinner

a) Marginal pmf

X

f(x)

12

0.2

15

0.5

20

0.3

Y

f(y)

12

0.1

15

0.35

20

0.55

b) P(X ≤ 15, Y ≤ 15) = 0.25

c) fX(12)fY(12) = 0.02 ≠ fX,Y(12,12), thus this is not independent.

d) E(X + Y) = 33.35

e) E(|X – Y|) = 3.85

22. Mean = 10, SD = 0.2

a) E(X1 + X2 + X3 + X4) = 40

b) Var(X1 + X2 + X3 + X4) = 0.16, SD(X1 + X2 + X3 + X4) = 0.4

23. Same question as #17.

Math 3339

Test 1 Review

1. A highway engineer knows that his crew can lay 5 miles of highway on a clear day, 2

miles on a rainy day, and only 1 mile on a snowy day. Suppose the probabilities are as

follows: A clear day: .6, a rainy day: .3, a snowy day: .1. What are the mean and

variance?

2. The following is a stem-plot of the birth weights of male babies born to the smoking

group. The stems are in units of kg.

Stems Leaves

2

3,4,6,7,7,8,8,8,9

3

2,2,3,4,6,7,8,9

4

1,2,2,3,4,5,6

5

3,5,5,6

Find the median birth weight.

3. The heights in centimeters of 5 students are:

165, 175, 176, 159, 172

Find the sample median, sample mean and sample variance.

4. Newsweek in 1989 reported that 60% of young children have blood lead levels that could

impair their neurological development. Assuming a random sample from the population

of all school children at risk, find the probability that at least 5 children out of 10 in a

sample taken from a school may have a blood level that may impair development.

5. The test grades for a certain class were entered into a Minitab worksheet, and then

Descriptive Statistics were requested. The results were:

MTB> Describe ‘Grades’.

N

Grades

MEAN

MEDIAN TRMEAN STDEV SEMEAN

28

74.71

76.00

75.50 12.61

MIN

MAX

Q1

Q3

94.00

68.00

84.00

Grades 35.00

2.38

You happened to see, on a scrap of paper, that the lowest grades were 35, 57, 59, 60, …

but you don’t know what the other individual grades are. Nevertheless, a knowledgeable

user of statistics can tell a lot about the data set simply by studying the set of descriptive

statistics above.

a. Write a brief description of what the results in the box tell you about the

distribution of grades. Be sure to address:

i.

ii.

iii.

iv.

The general shape of the distribution

Unusual features, including possible outliers

The middle 50% of the data

Any significance in the difference between the mean and the median

b. Construct a boxplot for the test grades.

6. Suppose the probability that a company will be awarded a certain contract is .25, the

probability that it will be awarded a second contract is .21 and the probability that it will

get both contracts is .13. What is the probability that the company will win at least one of

the two contracts?

7. A psychologist interested in right-handedness versus left-handedness and in IQ scores

collected the following data from a random sample of 2000 high school students.

Right-handed

High IQ

190

Normal IQ 1710

Total

1900

Left-handed

10

90

100

Total

200

1800

2000

a. What is the probability that a student from this group has a high IQ?

b. What is the probability that a student has a high IQ given that she is left-handed?

c. Are high IQ and left-handed independent? Why or why not?

8. A VCR manufacturer receives 70% of his parts from factory F1 and the rest from factory

F2. Suppose that 3% of the output from F1 are defective while only 2% of the output

from F2 are defective.

a. What is the probability that a received part is defective?

b. If a randomly chosen part is defective, what is the probability it came from

factory F1?

9. A sports survey taken at THS shows that 48% of the respondents liked soccer, 66% liked

basketball and 38% liked hockey. Also, 30% liked soccer and basketball, 22% liked

basketball and hockey, and 28% liked soccer and hockey. Finally, 12% liked all three

sports.

a. Draw a Venn diagram to represent the given information.

b. What is the probability that a randomly selected student likes basketball or

hockey? Solve this by also using an appropriate formula.

c. What is the probability that a randomly selected student does not like any of these

sports?

10. Donald has ordered a computer and a desk from 2 different stores. Both items are to be

delivered Tuesday. The probability that the computer will be delivered before noon is .6

and the probability that the desk will be delivered before noon is .8. If the probability

that either the computer or the desk will be delivered before noon is .9, what is the

probability that both will be delivered before noon?

11. A distribution of grades in an introductory statistics class (where A = 4, B = 3, etc) is:

X

0

1

2

3

4

P(X) .10 .15 .30 .30 ?

a. Find P(X = 4)

b. Find P (1 ≤ X < 3)
c. Find the mean grade in this class.
d. Find the standard deviation for the class grades.
e. Find the lowest grade X 0 such that P ( X ≥ X 0 ) < 0.5
12. Suppose you have a distribution, X, with mean = 28 and standard deviation =2.1.
Define a new random variable Y = 2X + 1.
a. Find the mean of Y.
b. Find the variance of Y.
c. Find the standard deviation of Y.
d. Let W = X + X for X in the above problem. Find the variance of W.
13. An appliance store is offering a special price on a complete set of kitchen appliances
(refrigerator, oven, stove, dishwasher). A purchaser is offered a choice of manufacturer
for each component:
Refrigerator: Kenmore, GE, LG, Whirlpool
Oven: KitchenAid, Samsung, Frigidaire, Kenmore
Stove: Electrolux, Hotpoint, GE
Dishwasher: Bosch, Silhouette, Premier, Whirlpool
Use the product rules to answer the following questions:
a. In how many ways can one appliance of each type be selected?
b. In how many ways can appliances be selected if none is to be Kenmore?
c. If someone randomly chooses their appliances, what is the probability that at least
one Kenmore component is chosen?
14. Suppose that for events A and B,
a. Compute
b. Are events A and B independent?
15. Inventory for a manufacturer are produced at three different plants, 45% from plant 1,
30% from plant 2, and 25% from plant 3. In addition, each plant produces at different
levels of quality. Plant 1 produces 2% defectives, plant 2 produces 5% defectives, and
plant 3 produces 8% defectives.
a. What is the probability that an item is defective?
b. If an item from the inventory is found to be defective, what is the probability that
is was produced in plant 2?
16. An urn has 20 blue marbles and 15 red marbles in it. Determine the probability that if 5
marbles are selected, at least two will be blue.
17. Marie is getting married tomorrow, at an outdoor ceremony in the desert. In recent years,
it has rained only 5 days each year. Unfortunately, the weatherman has predicted rain for
tomorrow. When it actually rains, the weatherman correctly forecasts rain 90% of the
time. When it doesn't rain, he incorrectly forecasts rain 10% of the time. What is the
probability that it will rain on the day of Marie's wedding, given the weatherman
forecasts rain?
18. The grades of from a random sample of 9 students on a midterm exam (x) and on the final
exam (y) are as follows:
x
y
77
82
50
66
71
78
72
34
81
47
94
85
96
99
99
99
67
68
a) Draw a scatterplot of the relationship between the midterm exam scores and the final
exam scores.
b) From the scatterplot, determine the strength, direction and form of the relationship.
c) Give the correlation between the scores on the midterm and the scores on the final
exam. Describe what this number means.
d) Determine the least-squares regression line.
e) Estimate the final exam grade of a student who received a grade of 85 on the midterm
exam.
f) Determine the coefficient of determination. Describe what this number means.
19. An electronics firm claims that the proportion of defective units of a certain process is
5%. A buyer has a standard procedure of inspecting 15 units selected randomly from a
large lot. On a particular occasion, the bury found 5 items defective.
a) What is the probability of this occurrence, given that the claim of 5% defective is
correct?
b) What would be your reaction if you were the buyer?
c) Determine the expected number of defective items from 15 units selected, given that
the claim of 5% defective is correct.
20. Suppose that from a group of 9 men and 8 women, a committee of 5 people is to be
chosen.
a. What type of probability distribution is this?
b. What is the probability that the committee has exactly 3 men and 2 women?
21. A restaurant serves three fixed-price dinners costing $12, $15, and $20. For a randomly
selected couple dining at this restaurant, let X = the cost of the man’s dinner and Y = the
cost of the woman’s dinner. The joint pmf of X and Y is given in the following table:
P(x, y)
X
12
15
20
12
0.05
0.05
0
Y
15
0.05
0.10
0.20
20
0.10
0.35
0.10
c. Compute the marginal pmf’s of X and Y.
d. What is the probability that the man’s and the woman’s dinner cost at most $15
each?
e. Are X and Y independent? Justify your answer.
f. What is the expected value of the total cost of the dinner for the two people?
g. Suppose the when a couple opens fortune cookies at the conclusion of the meal,
they find the message “You will receive as a refund the difference between the
cost of the more expensive and the less expensive meal that you have chosen.”
How much does the restaurant expect to refund?
22. The weight of a randomly selected bag of corn chips coming off an assembly line is a
random variable with mean µ= 10 oz. and standard deviation σ= 0.2 oz. Suppose we pick
four bags at random assume that weight of each of the bags are independent.
a. What is the mean of the combined weight of these four bags?
b. What is the standard deviation of the combined weight of these four bags?
23. Marie is getting married tomorrow, at an outdoor ceremony in the desert. In recent years,
it has rained only 5 days each year. Unfortunately, the weatherman has predicted rain for
tomorrow. When it actually rains, the weatherman correctly forecasts rain 90% of the
time. When it doesn't rain, he incorrectly forecasts rain 10% of the time. What is the
probability that it will rain on the day of Marie's wedding, given the weatherman
forecasts rain?

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