math 3070 quiz 1hormann/quiz1to11.pdf · math 3070 quiz 2 name: no outside materials are allowed,...

Math 3070 Quiz 1

Name:

No outside materials are allowed, except pens, pensils or calculators. Keep your eyes

on your own paper. Don't forget to label all plots. Show all your work.

1. A market researcher wants to �nd out how many di�erent states of the U.S. a Utah resident inthe age between 20 and 45 has typically visited. He obtained the following sample:

18 5 18 11 22 13 18 15 16 2625 16 4 8 12 15 16 15 12 1618 13 22 16 12 8 15 20 21 9

a: What is the corresponding population? Is it concrete or conceptual? (1)

b: Make a stem-and-leaf plot with 6 stems. (3)

c: What are the frequency and the relative frequency of people having visited 15 states? (1)

d: What is the proportion of people who have visited 20 or more di�erent states? (1)

e: Regard the above data as continuous data and sketch a histogram in density scale with thefollowing class boundaries: 0, 10, 14, 17, 19, 28. Indicate the exact height of each bar! (4)

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Math 3070 Quiz 2

Name:

No outside materials are allowed, except pens, pensils or calculators. Keep your eyes

on your own paper. Don't forget to label all plots. Show all your work.

1. The following data are 10 monthly temperature averages (in �F ) of some city.

81 69 79 82 84 88 83 78 84 80

a: Specify the ordered sample. (1)

b: Compute the interquartile range q0:75 � q0:25. (3)

c: Compute the sample variance s2. (3)

d: Draw a boxplot indicating possible outliers. (3)

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Math 3070 Quiz 3

Name:

No outside materials are allowed, except pens, pencils or calculators. Keep your eyes

on your own paper. Show all your work. This sheet is two-sided!!!!

1. In addition to his white chalk a professor sometimes uses a red or a blue chalk. During asemester with 48 lectures a student observed that in 16 of his lectures he used a blue a chalk andin 24 a red chalk. Another student observed that in 12 of his lectures he used both colors.

a: What is the probability that in a randomly chosen lecture he used at least one of the coloredchalks? (2)

b: What is the probability that in a randomly chosen lecture he will use no colored chalks? (1)

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2. A manufacturer of memory cards knows that 4% of the cards have some defect after theproduction process. In order to guarantee high quality, each outgoing card is tested with a newspecial procedure.

However, in test runs it shows that 5% of the defect cards are not detected and that 2% of thefaultless cards are falsely classi�ed as defect.

a: What is the probability that a randomly chosen card is not defect and passes the test. (2)

b: What is the probability that a randomly chosen card passes the test. (2)

c: A card has passed the test. What is the probability that it is faultless. (3)

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Math 3070 Quiz 4

Name:



1. The system below has 5 components. The individual components work with probability 0.9,independent from each other. The system works, if component 5 works and if either components1,2 and 3 or component 4 work. Compute P (System works). (4)

��

PPPP

��1

��4 �

��

��2 ��3

@@��5

2. A random variable X can have the values 1, 2, 4 and 8. The probability mass function p(x) ofX is given by p(1) = 0:1, p(2) = 0:2, p(4) = 0:4. Compute p(8). (2)

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3. Assume a random number generator produces only the numbers 1 or 3, each with probability0:5. Two numbers are produced independently and then multiplied. Let X be the outcome of thismultiplication.

a: Describe the set of possible values of X. (2)

b: What is the probability mass function of X? (2)

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Math 3070 Quiz 5

Name:



1. We roll a (fair and six-sided) die 10 times. What is the probability that \3" is obtained at least2 times? (3)

2. Assume that 20 out of 30 pupils have done their homework. The teacher randomly chooses twopupils. What is the probability that both have done their homework? (2)

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3. Let X be the number of points of a student of a 3070 statistics class. We know that X has thefollowing distribution:

x 0 1 2 3 4 5 6 7 8 9 10p(x) 0 .05 .05 .05 .1 .1 .2 .2 .1 .1 .05

a: Compute the expected value of X. (2)b: Compute the variance of X. (3)

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Math 3070 Quiz 6

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1. Suppose that 0.1% of people have a certain disease. Consider a sample of 5000 people.

(a) What is the (approximate) probability that more than 3 people of this samplehave the disease? (3)

(b) What is the (approximate) probablility that none of the sampled peoplehas this disease? (2)

Hint: Use Poisson-Approximation.

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2. Deside whether the given function is a distribution function of a continuous random variable.Explain your answer!

� F (x) = 1� 1=px for x � 1, F (x) = 0 if x < 1. (1)

� F (x) = 1=(1 + x2). (1)

3. Assume that

F (x) =

8<:

0 if x < 0;x3 if x 2 [0; 1];1 if x > 1.

Compute the density function of F . (2)

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Math 3070 Quiz 7

Name:



1. Assume that the random variable X has the folloing distribution:

F (x) = e�1x2 if x > 0 and F (x) = 0 if x � 0.

a) Determine the value z0:8, i.e. the value that will be not exceeded with a probability of 80%. (2)

b) Compute the probability that X is observed in the interval [1; 3]. (2)

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2. Assume that F (x) = 1� x�2

2if x > 1 and F (x) = 0 if x � 1. Then F is a distribution function.

(a) Determine the density funcion of F . (2)

(b) Compute EX. (2)

(c) Let h(x) = minfx; 2g. Compute Eh(X). (2)

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Math 3070 Quiz 8

Name:



1. Assume that the discrete random vector (X;Y ) has the following joint distribution:

p(x; y) x1 2 3 4 5

0 0:02 0:1 0:05 0:05 0:03y 4 0:1 0 0 0:2 0:1

5 0 0:15 0:15 0:05 0

(a) Compute the marginal distributions of X and Y . (3)

(b) Compute Cov(X;Y ). (3)

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2. Someone claims to be a clairvoyant. We ip a coin 1000 times and ask the clairvoyant to predictwhether the outcome is head or tail. For 550 times he made the right forecast. If the person werea cheater: what is the probability that the number X of correct guesses is 550 or larger? (Hencewe assume that he makes the right guess with probability 1=2.) Use the normal approximationand give the approximate value using the table below. (4)

z 0 0:2 0:4 0:6 0:8 1 1:2 1:4 1:6 1:8 2 2:2 2:4 2:6 2:8 3 3:2�(z) :5 :58 :66 0:72 :79 :84 :88 :92 :95 :96 :98 :99 :992 :995 :997 :998 :999

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Math 3070 Quiz 9

Name:



1. Assume that the amount of snow falling in a certain region during the months December,January, February and March is described by the random variables X1; X2; X3 and X4 (in inches).We know the following about expected values and variances:

i 1 2 3 4EXi 40 30 40 20

Var(Xi) 160 60 140 90

(a) What is the expected value of the total amount of snow during the four month? (1)

(b) Assume that X1, X2, X3 and X4 are independent. What is the standard deviation of the totalamount of snow during these months? (1)

(c) If the random variables are independent and normally distributed. What is the probabilitythat the total amount of snow is less the 110 inches? (3)

Note: For (c) use the table below and the fact that �(�x) = 1� �(x).

z 0 0:2 0:4 0:6 0:8 1 1:2 1:4 1:6 1:8 2 2:2 2:4 2:6 2:8 3 3:2�(z) :5 :58 :66 0:72 :79 :84 :88 :92 :95 :96 :98 :99 :992 :995 :997 :998 :999

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2. Assume that X1; X2; : : : ; Xn are independent and identically distributed random variables witha common distribution function F (x) given by

F (x) =

(1� ��1

2x�� if x � 1,

0 else,

where � > 1. We would like to estimate �. Show that 2 �Xn is an unbiased estimator for �. (5)

Note: �Xn = 1

n(X1 + � � �+Xn).

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Math 3070 Quiz 10

Name:



1. Assume that the amount of soda in a certain type of beverage cans is normally distributed witha true standard deviation of 0.3 oz.

(a) The average content of a random sample of 16 cans was 18.2 oz. Compute a 95% con�denceinterval (CI) of the true average. (2)

(b) Compute a 95% lower bound for the true average. (2)

(c) How large sample size is necessary such that the length of a 95% CI of the true average is lessthan 0:05? (3)

Note: For (c) use the quantiles below.

q1�� 1:65 1:96� 0:05 :025

or if you learned from the book:z� 1:65 1:96� 0:05 :025

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2. Assume that X1; X2; : : : ; Xn are independent and identically distributed random variables witha common density function f(xj�) given by

f(xj�) = 1p2�

e�(x��)2

2 :

Compute the Maximum likelihood estimator for �. (4)

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Math 3070 Quiz 11

Name:



1. Assume that X, the amount of soda in a certain type of beverage cans, is normally distributedN(�; �2). Here is a sample of 9 cans:

10:1 9:8 9:6 9:7 10:1 9:5 9:9 9:5 9:6

(a) If � = 0:3, test H0 : � = 10 against HA : � < 10. Use signi�cance level � = 0:01. (2)

(b) Do the same as in (a) when � is not known (s = 0:217). (2)

(c) Compute the p-value in case of unknown �. (2)

Note: Use the tables on the back.

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2. We would like to �nd out whether a die is forged or not. In 120 games \6" was observed only15 times.

(a) Formulate H0 and HA and give a test statistic. (2)

(b) What is your answer, if we make a level � test with � = 0:1? (2)

Note: Use the tables on the back.

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