statistics final

A game: Flip two fair coins. If both are heads, you win $10. Otherwise you win nothing.What are your expected winnings in this game (assuming you pay nothing to play)?

$2.50

Suppose that we wish to know the mean weight of adult bald eagles living in a certain region. We take a random sample of 36 eagles, and find a sample mean of 12 lbs. with a standard deviation of 6 lbs. A 95% confidence interval for the population's mean weight (in lbs.) would be...

(10,14)

If you draw three cards at random from a deck, what's the probability of drawing three aces? (Note: Lest there be any misunderstanding, the cards are drawn here without replacement, meaning that once you've drawn a card, you don't put it back in the deck.)

.05769

Suppose that the histogram below represents students' scores on a calculus exam. If we pick a random student in this class, what is the probability that he or she earned an A on the test (i.e. a score of at least 90)?

.20

Flip nine fair coins. What is probability (to the nearest thousandth) of flipping exactly nine heads?

0.002

If you roll a die three times, what is the probability of rolling three ONES? (Give your answer as a decimal, rounded to the nearest thousandth. That is, rounded to three decimal places.)

0.004

Flip a coin 5 times. The probability of getting 5 tails is, to the nearest thousandth...

0.031

Approximately 1% of the population suffers from the terrible disease symptomitis. The test for symptomitis comes back positive 97% of the time when taken by those who have the disease; it comes back negative 79% of the time when taken by those who do not have the disease. If a randomly chosen person is given the test and the test comes back positive for symptomitis, what is the probability that he or she actually has the disease?

0.04

If you draw two cards at random from a deck without replacement, what is the probability of drawing two face cards? (The face cards are the Jacks, Queens, and Kings.)

0.05

Roll six fair dice. Find the probability of rolling exactly 3 sixes.

0.054

Suppose that the number of hours that dogs sleep per day is normally distributed, with a mean of 13, and a standard deviation of 2. The probability that a randomly selected dog sleeps more than 16 hours per day is approximately...

0.07

Suppose the adult inhabitants of Deimos (one of the two strangely-shaped Martian moons) have a mean height of 17 cm, with a standard deviation of 8 cm. Suppose, moreover, that these heights are not normally distributed. If we take a sample of 100 Deimosians, what is the probability that their mean height will be less than 16 cm?

0.11

Draw a card from a deck. To the nearest hundredth, what's the probability of drawing a red face card? ("face cards" = kings, queens, jacks)

0.12

Draw a card from a deck. To the nearest hundredth, what's the probability of drawing a card that is a black jack given that the card drawn is a face card?

0.17

If you don't know the answer to this question, and just pick one of the possible responses at random, what is the probability that you will be lucky and guess the correct answer?

0.2

Based on the histogram in the previous problem, if 60 or higher is considered a passing score, then what is the probability that a randomly chosen student earned an A, given that he or she passed the test?

0.22

Flip ten fair coins. What is the probability (to the nearest thousandth) of flipping exactly five heads?

0.246

Suppose that in your sock drawer you have 5 black socks, 3 white socks, and 3 blue socks. You grab two socks at random in the dark and put them on. What is the probability that you will end up wearing a matching pair of socks?

0.29

Suppose you have a biased coin that comes up heads 3/4 of the time. If you flip it six times, what's the probability that you'll get exactly four heads? (Rounded to the nearest hundredth)

0.30

If you roll two dice, what is the probability of rolling at least one six?

0.306

Skydiving from the top of a tall building (as opposed to a plane) is particularly risky. Suppose that, for any given skydiver on any given day, the probability of dying during an attempted dive from the Space Needle is 1/50. The great daredevil Mr. So-and-so declares that as a fitting end to his career, he is going to dive from the Space Needle every morning for the next 50 mornings, and then he will retire in glory. What is the probability that he will live to see his retirement?

0.364

If we select a random woman from the survey described in the previous problem, what is the probability that she is tattooed?

0.38

Suppose that Martian IQs are normally distributed with a mean of 212 and a standard deviation of 32. If you randomly select a Martian, what's the probability of selecting one whose IQ is greater than 222? (Round to the nearest hundredth.)

0.38

Suppose we have a test for a certain disease that affects 2% of the general population. When taken by people who have the disease, the test gives the correct result 99% of the time. When taken by people who do not have the disease, the test gives the correct result 97% of the time. Suppose a randomly chosen person is given the test, and the result is POSITIVE. What is the probability that this person has the disease? (Your answer should be given as a decimal rounded to the nearest thousandth.)

0.402

If the mean of a data set is 173 and the standard deviation is 32, what standard score corresponds to a raw score of 160? (Round to the nearest hundredth.)

0.41

If X is a normal random variable,What's the probability that X's value will be greater than the mean of X?

0.5

Draw a card from a deck. To the nearest hundredth, what's the probability of drawing a card that is either black OR a jack (or both)?

0.54

Suppose that a survey was conducted to study the prevalence of tattoos among adults in Olympia with the results shown in the table: Men Women Not tattooed 211 188 Tattooed 89 117 Pick a random person from this survey.What is the probability that the person is a woman or is tattooed?

0.65

Suppose that 10% of all students at SPSCC have passed a statistics class. If seven SPSCC students are chosen at random, what is the probability that at most one of them has passed statistics?

0.85

Suppose that of all Americans past the age of thirty, 30% have never been married, 50% have been married exactly once, 19% have been married exactly twice, and 1% have been married exactly 3 times. Find the expected value of the number of marriages for a randomly selected American over the age of 30.

0.91

Draw a card from a deck. Find the numerical value of P(Face Card | Queen). [Recall that the "face cards" are the Kings, Queens, and Jacks.]

1

Flip ten fair coins. Let X be the number of heads that come up. What is σX? (To the nearest hundredth)

1.58

A card is drawn at random from a deck. What is the probability that it is an ace, given that it is a spade?

1/13

Draw a card at random from a deck.What is P(King|Red) ?

1/13

Draw a card at random from a deck.What is P(Red|King) ?

1/2

Draw a card from a deck. Find the numerical value of P(Queen | Face Card). [Recall that the "face cards" are the Kings, Queens, and Jacks.]

1/3

If you draw a card at random from a deck, what is the probability of drawing the ace of spades?

1/52

If you've just rolled 3 consecutive ONES on a fair die, the probability of rolling a ONE on the next roll is...

1/6

Suppose an urn contains four balls: two are red and two are white.You reach in and take two of them at random.What is the probability that you get both of the red balls?

1/6

The graph below represents students' scores on an exam in a calculus class. If a score below 60 is considered a failure on the exam, what percentage of the class failed the test?

10%

In how many ways can the letters of the word "tapir" be arranged in a row?

120

(Still referring back to #1) To qualify as a member of SSOCS (The Serene Society Of Canine Sleepers), your dog must be in the top 4% of all dogs with respect to time spent asleep per day. How many hours per day must your dog sleep (at minimum) to qualify for this prestigious society?

16.50

If you draw a card at random from a deck, what is the probability of drawing an ace or a spade?

16/52

Consider the following data: 7, 1, 5, 3, 4.The standard deviation of this data set is...

2

Now I'll select a group of four students from a different class of 40, and give one student in the group an A, another a B, another a C, and another an F. In how many different ways can I do this?

2,193,360

Suppose the names of 10 tapirs are written on slips of paper, and these slips of paper are put into a hat. We shake up the hat and draw out four slips. Each of the four tapirs whose name is selected will be given an apple. How many different quartets of tapirs might be selected in this way?

210

As in the previous problem, suppose that Martian IQs are normally distributed with a mean of 212 and a standard deviation of 32. What is the minimum IQ needed for a Martian to be in the top fifth of all Martians, as ranked by IQ scores? (Note that an IQ score must be a whole number.)

239

Consider a combination lock that opens only when one enters the correct sequence of three numbers. The numbers one can enter are each between 0 and 59 (inclusively), and numbers can be repeated (i.e. 32-14-32 is a legitimate sequence). Assuming one can check a sequence every 4 seconds, how long would it take to try every possible sequence?

240 hours

When you place an order at a certain sandwich shop, you must select one type of bread (from among 5 options), one type of meat (from among 5 options), two types of cheese (from among 7 options), and four types of vegetables (from among 12 options). How many different sandwiches can be ordered?

259,875

If X is a random variable with the distribution below, what is the value of σx? X 2 3 10 Prob 0.1 0.6 0.3

3.3

Roll a die, and let X be the result. What's the expected value of X?

3.5

Suppose that 30 randomly selected 5th graders are asked to rate on a scale of 1 to 5 how important it is to do well in school, with 1 representing "not at all important" and 5 representing "very important". The results of this study are displayed in the following graph. Let X be the rating given by a randomly chosen student from this sample. What is E(X)?

3.77

If we pick a random couple with three children, what is the probability that they will have two girls and a boy (assuming that each child is equally likely to be a boy or a girl)?

3/8

The "Normal Rule" tells us that within 3, 2, and 1 SDs of a normal distribution's mean, we we see approximately 99.7%, 95%, and 68% of the data respectively. Let's extend this. Within half a SD of a normal distribution's mean, what percentage of the data should we expect to see? (Rounded to the nearest percentage point.)

38%

Flip eight fair coins. Let X be the number of heads that come up. What is E(X)?

4

Once again, suppose you have a biased coin that comes up heads 3/4 of the time. What is the expected value of the number of heads obtained in six flips?

4.5

Suppose we put the letters S, C, R, A, M, B, L, E in a hat. We mix these around, and then pull them out one at a time at random, creating a (probably meaningless) "word" in the process. (For example, we might end up with "MASCRELB", "LCSMABRE", or even "MRCABLES".) How many different "words" can be generated this way?

40,320

Suppose we have a special die, whose six faces have the following numbers on them: 3, 3, 3, 7, 10, 10. Let X be the result we obtain if we roll this die once.What is the expected value of X?

6

To the nearest percent, what is the probability of flipping four consecutive heads on a fair coin?

6%

Suppose I select a random group of five students (from a class of 40) and will give each student in the group an automatic A in the class. How many different groups of five can I choose?

658,008

A statistics teacher gives an exam. Before returning the graded exams in the next class, he tells his students that the mean score on the exam was a 60 with a standard deviation of 16. On the exams that he returns, he doesn't write the actual test scores; instead he writes the standardized scores. Nancy receives her test back, and finds that her standardized score is 3/4. What is Nancy's actual test score?

72

Alice has 2 tapirs, Ben has 5 tapirs, and Carlos has 8 tapirs. Each of these three friends agrees to bring one of his or her tapirs to the park to have a tapir party. How many different tapir trios are possible at the party?

80

A medical researcher has 12 volunteers for testing a new drug. She will select six of them for her study. (All six will receive the same treatment.) How many different groups of six can she select?

924

(Continued from the previous question) Out of every 10,000 women in The Grand Duchy of Fenwick, approximately how many would you expect to be over 5 feet tall?

9452

The heights of women in The Grand Duchy of Fenwick are normally distributed with a mean of 64 inches and a SD of 2.5 inches. Approximately what percentage of women in The Grand Duchy of Fenwick are between 59 and 69.25 inches?

96%

Suppose that in a certain class, you will have five tests. Your lowest test score will be dropped, and the mean of the remaining four tests will then determine your final grade for the class. If their mean turns out to be 93 or higher, you will get an A in the class. Suppose that your first four test scores were as follows: 89, 76, 95, 89. What is the lowest score you can get on the fifth test if you are to earn an A in the class?

99

Suppose that in a statistics class of 40 students, the mean score on an exam was 80 and the standard deviation was 10. Which of the following is necessarily true?

All of the above are necessarily true.

Suppose a class of 400 students take a test. Their mean score is 70, with a standard deviation of 12, and the scores are normally distributed. Which of the following is necessarily true?

Approximately 272 students had scores between 58 and 82.

IQ scores are normally distributed with a mean of 100 and standard deviation of 15. So...

Approximately 68% of IQ scores are between 85 and 115.

The heights of women in the US are normally distributed. Suppose that the mean is 64 inches with a standard deviation of 2.5 inches. Which of the following statements is true...

Approximately 68% of women in the US are between 61.5 and 66.5 inches.

Suppose that a class of 40 students takes a test. Analyzing their scores, the professor finds that the MEAN = 70 and SD = 12. Which of the following statements is definitely true?

At least 30 students in the class had scores between 46 and 94.

Suppose that a study reveals that among all Americans, the mean number of peanuts eaten last year was 293 with a standard deviation of 84. Chebyshev's Theorem would then imply that...

At least 75% of Americans ate between 125 and 461 peanuts last year.

Suppose that in a certain company, the employees receive a mean of 10 days of paid vacation per year. Suppose further that a union pressures the company's manager to increase this mean. On reflection, the manager realizes he can satisfy the union demands by firing 5 of his employees. To accomplish this, which of the following would the manager have to do?

Fire ten of his employees who currently get fewer than the mean number of paid vacation days.

Suppose that you have just flipped five consecutive tails on a fair coin. On the next flip...

Heads and tails are equally likely.

A random variable's expected value is its...

Long-run mean

Either the sun will explode tomorrow, or it won't. Obviously, one - and only one - of these two possible outcomes will occur. From this fact, we may deduce that...

None of the above.

Regarding the effect that outliers have on data...

Outliers tend to affect the mean more than the median.

We are going to play a game: you'll roll many dice, and if at least 50% of the dice turn out to be sixes, then you win the game. Otherwise I win. You, however, are allowed to select the number of dice that you will roll: you can choose to roll 5 dice, 120 dice, or 1000 dice. Which choice should you make so as to maximize your chance of winning the game?

Roll 5 dice

(Continued from the previous problem) Nancy's friend Mary thinks to herself, "This test was pretty tough. I'll bet half the class had standard scores below -2." What can we say about Mary's idea? [Hint: What would Pafnuty Chebyshev say?]

She's definitely wrong.

Approximately 68% of women in the US are between 61.5 and 66.5 inches.

The mean

Suppose the owner of a company gives all his employees a flat raise of $200 per month. As a result...

The mean salary will increase, but the standard deviation will stay the same.

Which of the following is not a binomial random variable?

The number of times you must a roll a die until you get a six.

Based on the graph in Question 2, how many students in the class had scores of at least 95?

We don't have enough information to answer this question.

Which of the following could be a valid probability distribution for some random variable X?

X -4 1.5 10 Prob 0.6 0.1 0.3

Suppose a random variable X generates a fixed number (n) of values.Let X¯ be the mean of these values. Which of the following implies that X¯ must be a normal random variable?

X itself is normal

Suppose that in a data set containing 1000 numbers, the mean is 55 and the standard deviation is 15. Chebyshev's Theorem guarantees that...

at least 750 of the numbers lie between 25 and 85.

Suppose the mean height for an American man is 70 inches (that is, 5'10") with a standard deviation of 3 inches. Assuming that the heights of men are normally distributed, we can conclude that approximately 95% of American men are...

between 5'4" and 6'4".

What American city is famous for Boston Baked Beans?

boston

Another game: You pay $10 to play. After paying, you roll a die.If you roll a 1 or a 6, you are given $14 back. (Your original $10 plus $4 more.)If you roll anything else, you get nothing back. (You've lost your ten dollars.) If you play this game 500 times, what is your total expected profit (or loss)?

none of the above.

Either it will snow tomorrow or it won't. Obviously, one - and only one - of these two possible outcomes will occur. Consequently, we can conclude that...

none of the above.

(Continued from #1) A dog who sleeps 14 hours a day is at what percentile?

the 69th percentile

The expected value of a random variable X is...

the long-run mean of X (i.e. if we let X take on N different values, the mean of all these values gets closer and closer to E(X) as N gets larger and larger.)

"Think about how stupid the average person is.And then realize that half of 'em are stupider than that!" - George Carlin The average to which Carlin implicitly refers is...

the median

Of the following random variables, which is a binomial random variable?

the number of heads flipped if we flip a coin 42 times

The standard score of a particular value in a data set is defined to be...

the number of standard deviations it lies above (or below, if it is negative) the mean of the data.

What color was George Washington's white horse?

white