Math 234Homework 1

Part 1: Read Ch. 1-2, Walpole-Myers-Myers-Ye

Part 2: Solve the problems below

Due: February 7, 2021, 11:59 PM

1. The following measurements were recorded for the daily study time, in hours, of a group of college

students the day before an exam. Assume that the measurements are a simple random sample.

3.4

2.5

4.8

2.9

3.6

5.6

3.7

2.8

4.4

4

2.8

3.3

5.2

3

4.8

(a) What is the sample size for the above sample?

(b) Calculate the sample mean for these data.

(c) Calculate the sample median.

(d) Plot the data by way of a dot plot.

(e) Compute the 20% trimmed mean for the above data set.

(f) Is the sample mean for these data more-or-less descriptive as a center of location than the

trimmed mean?

(g) Find the sample variance and standard deviation.

2. Twenty delivery drivers between the ages of 30 and 40 participated in a study to evaluate the effect of

using the stairs instead of the elevator on weight. Ten were randomly selected to be a control group (and

used an elevator whenever possible), and ten others were assigned to only use stairs (treatment group) for

a period of 6 months. The following data show the reduction in weight (weight gain is marked by a

negative sign) experienced during the testing period for the 20 subjects:

Control

7

3

-4

14

2

Treatment

5

22

-7

9

5

-6

5

9

4

4

12

37

5

3

3

(a) Do a dot plot of the data for both groups on the same graph.

(b) Compute the mean, median, and 10% trimmed mean for both groups.

(c) Explain why the difference in means suggests one conclusion about the effect of using the stairs

on weight, while the difference in medians or trimmed means suggests a different conclusion.

(d) Compute the sample variance and the sample standard deviation for both control and treatment

groups.

3. A study of the effects of soda drinking on sleep patterns is conducted. The measure observed is the

time, in minutes, that it takes to fall asleep.

These data are obtained:

Soda-drinkers

69.3

60.2

56

43.8

22.1

23.2

53.2

47.6

48.1

34.4

52.7

13.8

(a)

(b)

(c)

(d)

Non-soda-drinkers

30.6

31.8

41.6

36

37.9

13.9

28.6

25.1

26.4

29.8

28.4

38.5

34.9

30.2

21.1

Find the sample mean for each group.

Find the sample standard deviation for each group.

Make a dot plot of the data sets A and B on the same line.

Comment on what kind of impact drinking soda appears to have on the time required to fall

asleep.

4. Below are the lifetimes, in hours, of fifty 40-watt, 110-volt internally frosted incandescent lamps, taken

from forced life tests. Construct a box plot for these data.

919

1196

1156

920

1170

929

1045

855

938

970

978

832

765

958

1217

1085

702

923

785

1126

936

918

948

1067

1092

1162

950

905

972

1035

1195

1195

1340

1122

1237

956

1102

1157

1009

1157

1151

1009

902

1022

1333

811

896

958

1311

1037

5. A bearing is a machine element that constrains relative motion to only the desired motion and reduces

friction between moving parts. A study is done to determine the influence of the wear, y, of a bearing as a

function of the load, x, on the bearing. A designed experiment is used for this study. Three levels of load

were used, 700 lb, 1000 lb, and 1300 lb. Four specimens were used at each level, and the sample means

were, respectively, 210, 325, and 375.

𝑦1

𝑦2

𝑦3

𝑦4

700

145

105

260

330

𝑥

1000

250

195

375

480

1300

150

180

420

750

𝑦̅1 = 210

𝑦̅2 = 325

𝑦̅3 = 375

(a) Plot average wear against load.

(b) From the plot in (a), does it appear as if a relation- ship exists between wear and load?

(c) Suppose we look at the individual wear values for each of the four specimens at each load

level (see the data that follow). Plot the wear results for all specimens against the three load

values.

(d) From your plot in (c), does it appear as if a clear relationship exists? If your answer is

different from that in (b), explain why.

Math 234

Homework 2

Part 1: Read Ch. 3-4, Walpole-Myers-Myers-Ye

Part 2: Solve the problems below

Due: February 20, 2021, 11:59 PM

1. Consider the density function

𝑓(𝑥) = {𝑘 √𝑥,

0,

0 < 𝑥 < 1,
elsewhere.
a) Evaluate 𝑘.
b) Find 𝐹(𝑥) and use it to evaluate 𝑃(0.3 < 𝑋 < 0.6).
2. From a box containing 4 dimes and 2 nickels, 3 coins are selected at random without replacement. Find
the probability distribution for the total T of the 3 coins. Express the probability distribution graphically
as a probability histogram.
3. Suppose it is known from large amounts of historical data that X, the number of cars that arrive at a
specific intersection during a 20-second time period, is characterized by the following discrete probability
function:
𝑓(𝑥) = 𝑒 −6
6𝑥
, for 𝑥 = 0,1,2, …
𝑥!
a) Find the probability that in a specific 20-second time period, more than 8 cars arrive at the
intersection.
b) Find the probability that only 2 cars arrive.
4. Determine the values of c so that the following functions represent joint probability distributions of the
random variables X and Y.
a) 𝑓(𝑥, 𝑦) = 𝑐𝑥𝑦, for 𝑥 = 1,2,3; 𝑦 = 1,2,3;
b) 𝑓(𝑥, 𝑦) = 𝑐|𝑥 − 𝑦|, for 𝑥 = −2,0,2; 𝑦 = −2,3.
5. Let X denote the diameter of an armored electric cable and Y denote the diameter of the ceramic mold
that makes the cable. Both X and Y are scaled so that they range between 0 and 1. Suppose that X and Y
have the joint density
1
,
𝑓(𝑥) = {𝑦
0,
0 < 𝑥 < 𝑦 < 1,
elsewhere.
1
Find 𝑃 (𝑋 + 𝑌 > 2).

6. A chemical system that results from a chemical reaction has two important components among others

in a blend. The joint distribution describing the pro- portions 𝑋1 and 𝑋2 of these two components is given

by

2,

𝑓(𝑥1 , 𝑥2 ) = {

0,

a)

b)

c)

d)

0 < 𝑥1 < 𝑥2 < 1,
elsewhere.
Give the marginal distribution of 𝑋1 .
Give the marginal distribution of 𝑋2 .
What is the probability that component proportions produce the results 𝑋1 < 0.2 and 𝑋2 > 0.5?

Give the conditional distribution 𝑓𝑋1 |𝑋2 (𝑥1 |𝑥2 ).

7. A large industrial firm purchases several new word processors at the end of each year, the exact

number depending on the frequency of repairs in the previous year. Suppose that the number of word

processors, X, purchased each year has the following probability distribution:

If the cost of the desired model is $1200 per unit and at the end of the year a refund of 50𝑋 2 dollars will

be issued, how much can this firm expect to spend on new word processors during this year?

8. Suppose that the probabilities are 0.4, 0.3, 0.2, and 0.1, respectively, that 0, 1, 2, or 3 power failures

will strike a certain subdivision in any given year. Find the mean and variance of the random variable X

representing the number of power failures striking this sub-division.

9. Suppose that a grocery store purchases 5 cartons of skim milk at the wholesale price of $1.20 per

carton and retails the milk at $1.65 per carton. After the expiration date, the unsold milk is removed from

the shelf and the grocer receives a credit from the distributor equal to three-fourths of the wholesale price.

If the probability distribution of the random variable X, the number of cartons that are sold from this lot,

is

find the expected profit.

10. Let X represent the number that occurs when a red die is tossed and Y the number that occurs when a

green die is tossed. Find

a) E(X+Y);

b) E(X-Y);

c) E(XY).

11. Let X be a random variable with the following probability distribution:

Find 𝐸(𝑋) and 𝐸(𝑋 2 ) and then, using these values, evaluate 𝐸[(2𝑋 + 1)2 ].

12. An electrical firm manufactures a 100-watt light bulb, which, according to specifications written on

the package, has a mean life of 900 hours with a standard deviation of 50 hours. At most, what percentage

of the bulbs fail to last even 700 hours? Assume that the distribution is symmetric about the mean.

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