Question 1

A student answers 26 questions correctly.Is that enough to convince you that he is not merely guessing? Explain.

Accepted Answer

A) No; if the student were guessing,we would expect him to answer 15 questions correctly with a standard deviation of 11.25.26 is 0.98 standard deviations above the expected value.That would not be an unusual result. 
B) Yes; if the student were guessing,we would expect him to answer 15 questions correctly with a standard deviation of 3.10.26 is 3.5 standard deviations above the expected value.That would be an unusual result. 
C) Yes; if the student were guessing,we would expect him to answer 15 questions correctly with a standard deviation of 3.35.26 is 3.3 standard deviations above the expected value.That would be an unusual result. 
D) Yes; if the student were guessing,we would expect him to answer 15 questions correctly with a standard deviation of 3.87.26 is 2.8 standard deviations above the expected value.That would be an unusual result. 
E) No; if the student were guessing,we would expect him to answer 15 questions correctly with a standard deviation of 3.35.26 is 3.3 standard deviations above the expected value.That would not be an unusual result. 
A) No; if the student were guessing,we would expect him to answer 15 questions correctly with a standard deviation of 11.25.26 is 0.98 standard deviations above the expected value.That would not be an unusual result. 
B) Yes; if the student were guessing,we would expect him to answer 15 questions correctly with a standard deviation of 3.10.26 is 3.5 standard deviations above the expected value.That would be an unusual result. 
C) Yes; if the student were guessing,we would expect him to answer 15 questions correctly with a standard deviation of 3.35.26 is 3.3 standard deviations above the expected value.That would be an unusual result. 
D) Yes; if the student were guessing,we would expect him to answer 15 questions correctly with a standard deviation of 3.87.26 is 2.8 standard deviations above the expected value.That would be an unusual result. 
E) No; if the student were guessing,we would expect him to answer 15 questions correctly with a standard deviation of 3.35.26 is 3.3 standard deviations above the expected value.That would not be an unusual result.

Question 2

The amount of money that Maria earns in a week is a random variable with a mean of $930 and a standard deviation of $30.The amount of money that Elena earns in a week is a random variable with a mean of $780 and a standard deviation of $15.If the difference between Maria's weekly income and Elena's weekly income can be described be a Normal model,what is the probability that Maria's weekly income is at least $223.79 more than Elena's weekly income? (In other words,what is the probability that the difference $\text { M-E }$ is at least $223.79?)Assume that Maria's earnings are independent of Elena's earnings.

Accepted Answer

A) 0.018 
B) 0.986 
C) 0.011 
D) 0.982 
E) 0.014 
A) 0.018 
B) 0.986 
C) 0.011 
D) 0.982 
E) 0.014

Question 3

An archer is able to hit the bull's-eye 48% of the time.If she shoots 8 arrows,what is the probability that she gets exactly 4 bull's-eyes? Assume each shot is independent of the others.

Accepted Answer

A) 0.0531 
B) 0.2717 
C) 0.0039 
D) 0.2355 
E) 0.7283 
A) 0.0531 
B) 0.2717 
C) 0.0039 
D) 0.2355 
E) 0.7283

Question 4

For a certain type of fabric,the average number of defects in each square foot of fabric is 0.9.Find the probability that a randomly selected square foot of the fabric will contain more than one defect.

Accepted Answer

A) 0.5934 
B) 0.2275 
C) 0.6341 
D) 0.7725 
E) 0.1647 
A) 0.5934 
B) 0.2275 
C) 0.6341 
D) 0.7725 
E) 0.1647

Question 5

The amount of money that Maria earns in a week is a random variable with a mean of $980 and a standard deviation of $25.The amount of money that Elena earns in a week is a random variable with a mean of $800 and a standard deviation of $10. You would like to use a Normal model to determine the probability that Maria's weekly income is at least $233.85 more than Elena's weekly income (the probability that the difference $M - E$ is at least $233.85).
Which of the following assumptions are needed?
A: Maria's weekly earnings are independent of Elena's weekly earnings.
B: Maria's weekly earnings and Elena's weekly earnings follow a Normal model.
C: Maria's weekly earnings are greater than Elena's weekly earnings

Accepted Answer

A) A and C 
B) A only 
C) B only 
D) A,B,and C 
E) A and B 
A) A and C 
B) A only 
C) B only 
D) A,B,and C 
E) A and B

Question 6

In the town of Blue Valley,7% of female college students suffer from manic-depressive illness.If 170 of the female students are selected at random,what is the mean number who suffer from manic-depressive illness?

Accepted Answer

A) 11.07 
B) 158.1 
C) 85 
D) 3.33 
E) 11.9 
A) 11.07 
B) 158.1 
C) 85 
D) 3.33 
E) 11.9

Question 7

Consider a game that consists of dealing out a hand of two random cards from a deck of four cards.The deck contains the Ace of Spades (As),the Ace of Hearts (Ah),the King of Spades (Ks)and the 9 of Hearts (9h).Aces count as 1 or 11.Kings count as 10.You are interested in the total count of the two cards,with a maximum count of 21 (that is,AsAh = 12).Let X be the sum of the two cards.Find the standard deviation of X.

Accepted Answer

A) 0.92 
B) 3.54 
C) 0.96 
D) 4.72 
E) 22.25 
A) 0.92 
B) 3.54 
C) 0.96 
D) 4.72 
E) 22.25

Question 8

Given independent random variables with means and standard deviations as shown,find the mean and standard deviation of the variable X + Y.Round to two decimal places if necessary. $$\begin{array} { l | l l } 
& \text { Mean } & \text { SD } \
\hline \mathrm { X } & 40 & 4 \
\mathrm { Y } & 70 & 7
\end{array}$$

Accepted Answer

A) ? = 2800,? = 11 
B) ? = 110,? = 65 
C) ? = 110,? = 8.06 
D) ? = 2800,? = 8.06 
E) ? = 110,? = 11 
A) ? = 2800,? = 11 
B) ? = 110,? = 65 
C) ? = 110,? = 8.06 
D) ? = 2800,? = 8.06 
E) ? = 110,? = 11

Question 9

Suppose the probability of a major earthquake on a given day is 1 out of 10,000.What is the expected number of earthquakes in the next 2000 days?

Accepted Answer

A)  $\frac { 1 } { 10,000 }$ 
B) 5 
C) 20 
D)  $\frac { 1 } { 2000 }$ 
E)  $\frac { 1 } { 5 }$ 
A)  $\frac { 1 } { 10,000 }$ 
B) 5 
C) 20 
D)  $\frac { 1 } { 2000 }$ 
E)  $\frac { 1 } { 5 }$

Question 10

The accompanying table describes the probability distribution for the number of adults in a certain town (among 4 randomly selected adults)who have a college degree. $$\begin{array} { c | c } 
\mathrm { x } & \mathrm { P } ( \mathrm { x } ) \
\hline 0 & 0.4096 \
1 & 0.4096 \
2 & 0.1536 \
3 & 0.0256 \
4 & 0.0016
\end{array}$$

Accepted Answer

A) 1.21 
B) 0.70 
C) 2.00 
D) 0.95 
E) 0.80 
A) 1.21 
B) 0.70 
C) 2.00 
D) 0.95 
E) 0.80

Question 11

In a certain town,8% of female college students suffer from manic-depressive illness.If 230 of the female students are selected at random,what is the standard deviation of the number who suffer from manic-depressive illness?

Accepted Answer

A) 4.11 
B) 4.29 
C) 211.6 
D) 18.4 
E) 16.93 
A) 4.11 
B) 4.29 
C) 211.6 
D) 18.4 
E) 16.93

Question 12

We draw a card from a deck 6 times (replacing the card after each draw)and get 3 kings.How likely is this?

Accepted Answer

A) Yes. 
B) No.More than two outcomes are possible. 
C) No.The chance of getting a king changes as cards are drawn. 
D) No.6 is more than 10% of 52. 
E) No.The draws are not independent of each other. 
A) Yes. 
B) No.More than two outcomes are possible. 
C) No.The chance of getting a king changes as cards are drawn. 
D) No.6 is more than 10% of 52. 
E) No.The draws are not independent of each other.

Question 13

Suppose you pay $2.00 to roll a fair die with the understanding that you will get back $$\$ 4.00$$ for rolling a 6 or a 3,nothing otherwise.What is the expected amount you win?

Accepted Answer

A) $2.00 
B) -$0.67 
C) $4.00 
D) -$2.00 
E) $0.00 
A) $2.00 
B) -$0.67 
C) $4.00 
D) -$2.00 
E) $0.00

Question 14

The probability that a car will have a flat tire while driving through a certain tunnel is 0.00006.Use the Poisson approximation to the binomial distribution to find the probability that among 13,000 cars passing through this tunnel,at most two will have a flat tire.

Accepted Answer

A) 0.1394 
B) 0.1840 
C) 0.9554 
D) 0.8160 
E) 0.8606 
A) 0.1394 
B) 0.1840 
C) 0.9554 
D) 0.8160 
E) 0.8606

Question 15

A laboratory worker finds that 1.6% of his blood samples test positive for the HIV virus.In a random sample of 130 blood tests,what is the standard deviation of the number that test positive for the HIV virus?

Accepted Answer

A) 2.04672 
B) 11.31 
C) 2.08 
D) 1.44 
E) 1.43 
A) 2.04672 
B) 11.31 
C) 2.08 
D) 1.44 
E) 1.43

Question 16

A basketball player has made 70% of his foul shots during the season.If he shoots 6 foul shots in tonight's game,what is the probability that he doesn't make all of the shots?

Accepted Answer

A) 0.3 
B) 0.0007 
C) 0.18 
D) 0.1176 
E) 0.8824 
A) 0.3 
B) 0.0007 
C) 0.18 
D) 0.1176 
E) 0.8824

Question 17

The amount of money that Jon can save after working for a summer is a random variable S with a mean of $\mu _ { \mathrm { s } }$ and a standard deviation of $\sigma _ { s }$ .After saving this money Jon plans to go on a trip to India.He will change his money into Rupees at an exchange rate of 43 Rupees to one Dollar.This money he will bring to India.When he arrives in India he will buy a used motorbike.The price in India of a motorbike of the type he wants is a random variable B with a mean of $\mu \mathrm { b }$ Rupees and a standard deviation of $\sigma b$ Rupees. The amount of money Jon will have left (in Rupees)after changing his savings into Rupees and buying a motorbike in India is a random variable P which can be expressed in terms of S and B as $P = 43 S - B$ Find expressions for the mean and variance of the random variable P.Assume that Jon's savings and the price of the bike are independent.

Accepted Answer

A) mean = 43 $\mu _ { \mathrm { s } }$ - $\mu \mathrm { b }$ ,variance = 1849 $\sigma _ { s } ^ { 2 }$ - $\sigma _ { b } ^ { 2 }$ 
B) mean = 43 $\mu _ { \mathrm { s } }$ - $\mu \mathrm { b }$ ,variance = 1849 $\sigma _ { s } ^ { 2 }$ + $\sigma _ { b } ^ { 2 }$ 
C) mean = 43 $\mu _ { \mathrm { s } }$ + $\mu \mathrm { b }$ ,variance = 1849 $\sigma _ { s } ^ { 2 }$ + $\sigma _ { b } ^ { 2 }$ 
D) mean = 43 $\mu _ { \mathrm { s } }$ + $\mu \mathrm { b }$ ,variance = 43 $\sigma _ { s } ^ { 2 }$ + $\sigma _ { b } ^ { 2 }$ 
E) mean = 43 $\mu _ { \mathrm { s } }$ - $\mu \mathrm { b }$ ,variance = 43 $\sigma _ { s } ^ { 2 }$ + $\sigma _ { b } ^ { 2 }$ 
A) mean = 43 $\mu _ { \mathrm { s } }$ - $\mu \mathrm { b }$ ,variance = 1849 $\sigma _ { s } ^ { 2 }$ - $\sigma _ { b } ^ { 2 }$ 
B) mean = 43 $\mu _ { \mathrm { s } }$ - $\mu \mathrm { b }$ ,variance = 1849 $\sigma _ { s } ^ { 2 }$ + $\sigma _ { b } ^ { 2 }$ 
C) mean = 43 $\mu _ { \mathrm { s } }$ + $\mu \mathrm { b }$ ,variance = 1849 $\sigma _ { s } ^ { 2 }$ + $\sigma _ { b } ^ { 2 }$ 
D) mean = 43 $\mu _ { \mathrm { s } }$ + $\mu \mathrm { b }$ ,variance = 43 $\sigma _ { s } ^ { 2 }$ + $\sigma _ { b } ^ { 2 }$ 
E) mean = 43 $\mu _ { \mathrm { s } }$ - $\mu \mathrm { b }$ ,variance = 43 $\sigma _ { s } ^ { 2 }$ + $\sigma _ { b } ^ { 2 }$

Question 18

Given independent random variables with means and standard deviations as shown,find the mean and standard deviation of the variable $$Y _ { 1 } + Y _ { 2 }$$ $$\begin{array} { c | c c } 
& \text { Mean } & \text { SD } \
\hline \mathrm { X } & 150 & 12 \
\mathrm { Y } & 200 & 18
\end{array}$$

Accepted Answer

A) ? = 350,? = 21.63 
B) ? = 350,? = 30 
C) ? = 400,? = 25.46 
D) ? = 400,? = 36 
E) ? = 400,? = 6.00 
A) ? = 350,? = 21.63 
B) ? = 350,? = 30 
C) ? = 400,? = 25.46 
D) ? = 400,? = 36 
E) ? = 400,? = 6.00

Question 19

Police estimate that 65% of drivers wear their seat belts.If they stop 4 drivers at random,what is the probability that none of them are wearing their seat belts?

Accepted Answer

A) 0.1785 
B) 0.26 
C) 0.0150 
D) 0.4 
E) 0.35 
A) 0.1785 
B) 0.26 
C) 0.0150 
D) 0.4 
E) 0.35

Question 20

A tennis player usually makes a successful first serve 74% of the time.She buys a new racket hoping that it will improve her success rate.During the first month of playing with her new racket she makes 402 successful first serves out of 500.Is this evidence that with the new racket her success rate has improved? In other words,is this an unusual result for her? Explain.

Accepted Answer

A) No; we would normally expect her to make 370 first serves with a standard deviation of 19.24.402 is 1.7 standard deviations above the expected value.That's not an unusual result. 
B) No; we would normally expect her to make 370 first serves with a standard deviation of 96.20.402 is 0.3 standard deviations above the expected value.That's not an unusual result. 
C) No; we would normally expect her to make 370 first serves with a standard deviation of 9.81.402 is 3.3 standard deviations above the expected value.That's not an unusual result. 
D) Yes; we would normally expect her to make 370 first serves with a standard deviation of 96.20.402 is 0.3 standard deviations above the expected value.That's an unusual result. 
E) Yes; we would normally expect her to make 370 first serves with a standard deviation of 9.81.402 is 3.3 standard deviations above the expected value.That's an unusual result. 
A) No; we would normally expect her to make 370 first serves with a standard deviation of 19.24.402 is 1.7 standard deviations above the expected value.That's not an unusual result. 
B) No; we would normally expect her to make 370 first serves with a standard deviation of 96.20.402 is 0.3 standard deviations above the expected value.That's not an unusual result. 
C) No; we would normally expect her to make 370 first serves with a standard deviation of 9.81.402 is 3.3 standard deviations above the expected value.That's not an unusual result. 
D) Yes; we would normally expect her to make 370 first serves with a standard deviation of 96.20.402 is 0.3 standard deviations above the expected value.That's an unusual result. 
E) Yes; we would normally expect her to make 370 first serves with a standard deviation of 9.81.402 is 3.3 standard deviations above the expected value.That's an unusual result.

Exam 14: Random Variables

A student answers 26 questions correctly.Is that enough to convince you that he is not merely guessing? Explain.

An archer is able to hit the bull's-eye 48% of the time.If she shoots 8 arrows,what is the probability that she gets exactly 4 bull's-eyes? Assume each shot is independent of the others.

For a certain type of fabric,the average number of defects in each square foot of fabric is 0.9.Find the probability that a randomly selected square foot of the fabric will contain more than one defect.

In the town of Blue Valley,7% of female college students suffer from manic-depressive illness.If 170 of the female students are selected at random,what is the mean number who suffer from manic-depressive illness?

Given independent random variables with means and standard deviations as shown,find the mean and standard deviation of the variable X + Y.Round to two decimal places if necessary. Mean SD 40 4 70 7

Suppose the probability of a major earthquake on a given day is 1 out of 10,000.What is the expected number of earthquakes in the next 2000 days?

The accompanying table describes the probability distribution for the number of adults in a certain town (among 4 randomly selected adults)who have a college degree. () 0 0.4096 1 0.4096 2 0.1536 3 0.0256 4 0.0016

In a certain town,8% of female college students suffer from manic-depressive illness.If 230 of the female students are selected at random,what is the standard deviation of the number who suffer from manic-depressive illness?

We draw a card from a deck 6 times (replacing the card after each draw)and get 3 kings.How likely is this?

Suppose you pay $2.00 to roll a fair die with the understanding that you will get back $4.00\$ 4.00$4.00 for rolling a 6 or a 3,nothing otherwise.What is the expected amount you win?

The probability that a car will have a flat tire while driving through a certain tunnel is 0.00006.Use the Poisson approximation to the binomial distribution to find the probability that among 13,000 cars passing through this tunnel,at most two will have a flat tire.

A laboratory worker finds that 1.6% of his blood samples test positive for the HIV virus.In a random sample of 130 blood tests,what is the standard deviation of the number that test positive for the HIV virus?

A basketball player has made 70% of his foul shots during the season.If he shoots 6 foul shots in tonight's game,what is the probability that he doesn't make all of the shots?

Given independent random variables with means and standard deviations as shown,find the mean and standard deviation of the variable Y1+Y2Y _ { 1 } + Y _ { 2 }Y1​+Y2​ Mean SD 150 12 200 18

Police estimate that 65% of drivers wear their seat belts.If they stop 4 drivers at random,what is the probability that none of them are wearing their seat belts?

Stats Starts Here

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The Standard Deviation As a Ruler and the Normal Model

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Testing Hypotheses About Proportions

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Suppose you pay $2.00 to roll a fair die with the understanding that you will get back $\$ 4.00$ for rolling a 6 or a 3,nothing otherwise.What is the expected amount you win?

Given independent random variables with means and standard deviations as shown,find the mean and standard deviation of the variable $Y _ { 1 } + Y _ { 2 }$ Mean SD 150 12 200 18