Learn Smart 4a

The probability that a newborn baby is a boy is slightly greater than 0.5. Assume the probability that a baby is a boy is 0.51. A couple has children one by one until the first baby girl is born. Which of the following statements is true about X?

If X is the number of children up to and including the first girl, then X ~ Geom(0.49).

A coin has probability 0.25 of coming up heads when it is tossed. Let X be the number of heads obtained when this coin is tossed 4 times. Select all that apply:

If the coin is tossed 4 times, each arrangement of 2 heads in 4 tosses has probability (0.252)(0.752). The possible arrangements of 2 heads in 4 tosses are {HHTT, HTTH, TTHH, HTHT, THTH, THHT}. P(X = 2) = (number of arrangements of 2 heads in 4 tosses)(0.252)(0.752).

For any Bernoulli trial, a Bernoulli random variable X can be defined. Which of the following are true statements about X?

If the experiment results in success, X = 1, Otherwise, X = 0.The probability mass function of X is given by P(0) = P(X = 0) = 1-p, where p is the probability of success, and p(1) = p(X =0) = 1-p, where p is the probability of success, and p(1) = p(X-1) = p

If X is a binomial random variable with parameters n = 7 and p = 0.8, and p(x) is the probability mass function of X, then ______

P(X = 7) = p(7) = 7!7!0!0.87 0.20. for x = 0, 1, 2,...,7, p(x) = 7!x!(7-x)!0.8x 0.27 - x. P(X = 2) = p(2) = 7!2!5!0.82 0.25.

If we observe X events occurring over a period of t hours in a Poisson process, the mean rate per hour λ is estimated as λˆ = X/t. Select all that apply.

The bias of λ̂ is 0, thus λ̂ is unbiased. The uncertainty in λ̂ can be estimated as σλ̂ = sqrtλ^/t.

An unbalanced four-sided pyramidal die has the numbers 1, 2, 3, 4, labeled on each side. Let p1, p2, p3, and p4 be the probabilities that a 1, 2, 3, or 4 comes up, respectively, when the die is rolled. Assume that p1 = 0.1, p2 = 0.2, p3 = 0.3, and p4 = 0.4. The die is tossed 10 times. Let Xi be the number of times die lands on the number i, i = 1, 2, 3, 4. Select all that apply.

The collection X1, X2, X3, X4 has a multinomial distribution with parameters n = 10, p1 = 0.1, p2 = 0.2, p3 = 0.3, and p4 = 0.4. The number of ordered ways that the die lands 2 times on 1, 3 times on 2, 4 times on 3, and 1 time on 4 in 10 tosses is 10!/(2!3!4!1!).

The proportions of M&Ms in different colors are: 25% blue, 14% brown,16% green, 20% orange, 12% red, and 13% yellow. You select 18 M&Ms. Let Xi, for i = 1, 2, 3, 4, 5, 6, be the number of blue, brown, green, orange, red, and yellow M&Ms, respectively. Select all that apply.

The collection X1,..., X6 has a multinomial distribution with n = 18, p1 = 0.25, p2 = 0.14, p3 = 0.16, p4 = 0.20, p5 = 0.12, and p6 = 0.13. The probability that there are 4 orange and 14 non-orange M&Ms is 18!14!4!(0.24)(0.814).

A computer program was run 1000 times to estimate the probability p that it crashes. The program crashed 41 times. Estimate p and find the uncertainty in the estimate.

The estimate of p is pˆ = 41/1000. The uncertainty in pˆ is (0.041)(0.959)/1000‾‾‾‾‾‾‾‾‾‾‾‾‾‾‾‾‾‾‾‾√.

The number of trees of a particular species in a large wooded area follows a Poisson distribution with mean λ trees per acre where λ is unknown. In 50 randomly selected one-acre plots of this wooded area, you count a total of 750 trees of this species. Estimate λ and find the uncertainty in the estimate. Select all that apply.

The estimate of λ is λˆ = 750/50=15. The uncertainty in λˆ is σλˆ = 1550‾‾‾√.

Five percent of the apples at a supermarket have blemishes. When purchasing apples for lunch for the entire week, you inspect apples one by one until you find 7 unblemished ones. Let X be the number of apples you inspect. Select all that apply.

The mean number of apples you inspect is 7/0.95. The variance of the number of apples you inspect is 7(0.05)/(0.95)2.

Five percent of all items manufactured by a company are defective. These items are packaged in boxes of 10. One of these boxes is randomly sampled, and X is the number of defective items in the box. Select all that apply.

The mean number of defective items in a box is 10(0.05). X ~ Bin(10, 0.05)

Sixty percent of people in a community are female. One person is randomly polled from this community. Let X = 1 if this person is female, and X = 0 otherwise. Select all that apply.

The mean of X is μX = 0.6. X ∼ Bernoulli(0.6).

A fair 6-sided die is rolled. Let X = 1 if an even number comes up, and X = 0 otherwise. Select all that apply:

The mean of X is μX = 1/2. X ~ Bernoulli(1/2).

A class has 18 female students and 12 male students. Ten students are randomly sampled. Let X be the number of female students in the sample. Select all that apply.

The mean of the Bin(10, 0.6) distribution is the same as the mean of the H(30, 18, 10) distribution. X has a hypergeometric distribution with parameters N = 30, R = 18, n = 10.

Out of 20 students in a statistics class, 12 are female and 8 are male. Five students are randomly sampled. Let X be the number of males in the sample. Select all that apply.

The minimum possible number of males in the sample is 0. The probability of having exactly 2 males in the sample is (82)(123)(205).

Which of the following must be true about a Poisson process that produces events that occur in time?

The number of events in disjoint intervals of time are independent. Events cannot occur simultaneously.

A coin has probability 0.25 of landing heads when it is tossed. Let X be the number of heads when this coin is tossed 4 times. Select all that apply:

The only possible arrangement of 4 heads in 4 tosses is HHHH. The number of arrangements of 4 heads in 4 tosses is 4!/4!0!.

A box contains 10 items. Two of them are defective, and eight are in good condition. Three items are randomly selected for inspection. Let X be the number of defective items in the sample. Select all that apply.

The probability of having 0 defective items in the sample is (20)(83)(103). The minimum possible number of defective items in the sample is 0.

A biased die with probability 0.35 of landing a 6 is rolled until the third 6 comes up. Let X be the number of rolls up to and including the third 6. Which of the following statements is true about X?

The smallest possible value of X is 3.

A biased coin has probability 0.4 of coming up tails. The coin is tossed 5 times. Let Yi = 1 if the coin comes up tails on the ith toss, for i = 1, 2, 3, 4, 5, and 0 if it comes up heads. Let X be the number of times this coin comes up tails when tossed 5 times. Select all that apply.

The sum X = Y1 + Y2 + Y3 + Y4 + Y5 is equal to the number of the Yi 's that have a value of 1. Each Yi has a Bernoulli distribution with parameter p = 0.4, i.e., each Yi ~ Bernoulli (0.4).

A student conducting a survey at a university found that 9 out of 50 randomly selected students were married. Estimate the proportion of married people at this university and find the uncertainty in the estimate.

The uncertainty in pˆ is 0.0543.The estimate of p is pˆ = 0.18.

A coin with probability 0.7 of coming up tails is tossed. Let X = 1 if a head comes up, and X = 0 otherwise. Select all that apply.

The variance of X is (0.3)(0.7) X bernoilli (0.3)

The probability that a computer program crashes each time it runs is 0.0001. Let X be the number of times this program runs up to and including the first time that it crashes. Select all that apply.

The variance of X is σ2X X ~ Geom(0.0001).

A committee of 5 people is selected at random from a group of 25 people that consists of 10 men and 15 women. Let X be the number of men on the committee, and let Y be the number of women on the committee. Select all that apply.

The variance of the number of women on the committee is 5(15/25)(10/25)(20/24). The mean number of men on the committee is 5(10)/25.

A biased coin has probability 0.9 of landing tails. The coin is tossed 12 times. Let X be the number of times this coin lands on tails. Select all that apply.

There are 12 independent Bernoulli trials in this experimentEach trial can be treated as having the same success probability p=0.9

A radioactive mass emits particles at a mean rate of 5 particles per hour. Let X be the number of particles emitted in a t-hour period. Select all that apply.

X has a Poisson(5t) distribution. The rate λ of particle emissions in any one-hour period can be estimated as λˆ = X/t.

A biased coin has probability 0.2 of landing tails. Each student in your class tosses this coin five times and records as X the number of tails in the five tosses. Select all that apply.

X has a binomial distribution with parameters n = 5 and p = 0.2, i.e., X ~ Bin(5, 0.2). The mean number of tails for each student is μX = 1.

A loaded die with probability 0.4 of landing 6 is tossed until the first 6 comes up. Let X be the number of tosses up to and including the first 6. Select all that apply.

X has a geometric distribution with parameter p = 0.4, i.e., X ~ Geom(0.4). The probability that the first 6 comes up on the fourth toss is (0.63)(0.4).

An urn contains 15 red balls and 5 green balls. Seven balls are randomly drawn from the urn. Let X be the number of red balls drawn. Select all that apply.

X has a hypergeometric distribution with parameters N = 20, R = 15, and n = 7. The probability that 5 red balls are drawn is (155)(52)(207).

A biased coin has probability 0.4 of landing tails when it is tossed 10 times. Let Y1 be the number of tosses up to and including the first tail, let Y2 be the number of tosses after the first tail up to and including the second tail, and let X be the total number of tosses up to and including the second tail. Select all that apply.

X has a negative binomial distribution with parameters r = 2 and p = 0.4. X = Y1 + Y2.

Sixty percent of the students at a large university are females. Students are sampled one by one. Let Y1 be the number of students sampled up to and including the first female, let Y2 be the number of students sampled after the first female up to and including the second female, let Y3 be the number of students sampled after the second female up to and including the third female, and let X = Y1 + Y2 + Y3. Select all that apply.

X has a negative binomial distribution with parameters r = 3 and p = 0.6. Each Yi ~ Geom(0.6).

A biased coin with probability 0.75 of landing tails is tossed until the fourth tail comes up. Let X be the number of tosses up to and including the fourth tail. Select all that apply.

X has a negative binomial distribution with parameters r = 4 and p = 0.75. The smallest possible value of X is 4.

Ten percent of the students at a large university are married. Students are polled one by one. Let X be the number of students polled until the fifth married student is selected. Which of the following statements is true about X?

X has a negative binomial distribution with parameters r = 5 and p = 0.1.

A university has 18,000 female students and 12,000 male students. One hundred students are randomly sampled. Let X be the number of female students in the sample. Select all that apply.

X is approximately Bin(100, 0.6). The mean of the Bin(100, 0.6) distribution is the same as the mean of the H(30,000, 18,000, 100) distribution. X has the hypergeometric distribution with parameters N = 30,000, R = 18,000, n = 100.

Out of hundreds of items produced at a factory, 1% are defective and need to be discarded, 6% are defective but can be refurbished, and 93% are in good condition. One of these items is randomly sampled. Let X = 1 if the item needs to be discarded, and X = 0 otherwise. Also, let Y = 1 if the item is defective (discardable or refurbishable), and Y = 0 otherwise. Select all that apply.

X ~ Bernoulli(0.01). The probability of success of Y is pY = 0.07.

Of the customers who purchase gas at a certain station, 45% purchase regular gas, 35% purchase plus, and 20% purchase supreme. Assume that no customer purchases more than one type of gas. Let X = 1 if the next customer purchases regular gas, and X = 0 otherwise. Also, let Y = 1 if the next customer purchases either regular or plus gas, and Y = 0 otherwise. Select all that apply:

X ~ Bernoulli(0.45) Y ~ Bernoulli(0.8)

Particles are suspended in a liquid medium at a concentration of 5 particles per mL. A large volume of the suspension is thoroughly agitated and 1 mL is sampled. Let X be the number of particles in the sample. Select all that apply.

X ~ Poisson(5). The probability that there are exactly 8 particles in the sample is e−5588!. μX = 5.

A drawer has 8 black pens and 6 red pens. Every day you randomly select 3 pens to take to school. Let X be the number of black pens you choose, and let Y be the number of red pens you choose. Select all that apply.

Y has the hypergeometric distribution with 6 members of the population categorized as successes. X has the hypergeometric distribution with 8 members of the population categorized as successes

Out of hundreds of items produced at a factory, 1% are defective and need to be discarded, 6% are defective but can be refurbished, and 93% are in good condition. One of these items is randomly sampled. Let X = 1 if the item needs to be discarded, and X = 0 otherwise. Also, let Y = 1 if the item is defective but can be refurbished, and Y = 0 otherwise. Select all that apply:

Y~Bernoulli (0.06)X+Y=Z, where Z~Bernoulli (0.07)

When tossing a fair six-sided die, let X = 1 if the dies comes up 3, and X = 0 otherwise. Also, let Y = 1 if the die comes up 3 or 5, and Y = 0 otherwise. Let Z = X + Y. Select all that apply.

Z = 1 if the die comes up 5. P(Z = 1) = 1/6.

When tossing a fair six-sided die, let X = 1 if the die comes up 3, and X = O otherwise. Also, let Y =1 if the die comes up 5, and Y=O otherwise. Finally, let Z=X+Y.

Z = 1 if the die lands on 3 or 5Z = 0 if the die lands on 1,2,4, or 6It is impossible that Z =2

The collection of random variables X1, X2,..., Xk has a multinomial distribution with parameters n and p1, p2 ,..., pk (i.e., X1, X2,..., Xk ~ MN(p1, ..., pk)). Select all that apply.

pi (i = 1, 2,..., k) is the probability of outcome i in each of the n multinomial trials. Each Xi has a binomial distribution with n trials and success probability pi. For each outcome i (i = 1, 2,..., k), Xi is the number of multinomial trials out of n that result in outcome i.

If X has a negative binomial distribution with parameters r and p, what is the mean of X?

r/p

The quantity np(1-p)‾‾‾‾‾‾‾‾√ is the _________ of a binomial random variable with parameters n and p.

standard deviation

If we observe X events occurring over a period of t hours in a Poisson process, the mean rate per hour is estimated as λˆ = X/t. We determine the precision of the estimate by computing the _________ of λˆ.

standard deviation uncertainty

A forest manager has pair of Landsat satellite images for each of 1,250 plots of land. A proportion p of these plots were logged within the past ten years. Each day she chooses a simple random sample of four plots, checks their image pairs, and categorizes them as logged or not by completing a spreadsheet like this:

the first value in the plot ID column matches The discrete distribution on {1, 2, 3, . . . , 1250}, The first value in the logged column matches,Bernoulli trial with success probability p The sum in tomorrow's spreadsheet matches , Bin(4,p) The mean in tomorrow's spreadsheet matches, None of the other distributions

Five percent of all items manufactured by a company are defective. These items are packaged in boxes of 40. Let X be the number of defective items in any randomly selected box. Then _______

the mean number of defective items per box is μX = 40(0.05). the variance of the number of defective items per box is 40(0.05)(0.95).

A coin has probability 0.25 of landing heads when it is tossed. Let X be the number of heads when this coin is tossed 4 times. Select all that apply:

the possible arrangement of 1 head in 4 tosses are (HTTT, THTT, TTHT, TTTH)P(x=2)=(number of arrangements of 2 heads in 4 tosses)(0.25^2)(0.75^2)The number of arrangements of 1 head in 4 tosses is 4!/1!(4-1)!

A coin was tossed 50 times, and in 15 of these tosses a head came up. The probability that a head comes up can be estimated as pˆ = 15/50.

true

Particles are suspended in a liquid medium at a concentration of 5 particles per mL. A large volume of the suspension is thoroughly agitated, and 3 mL is sampled. Let X be the number of particles in the sample.

μX = 15. The probability that there are exactly 8 particles in the sample is e−151588!.

If the sample proportion pˆ is used to estimate of the probability of success in a Bernoulli trial, then _________.

σpˆ= pˆ(1−pˆ)n‾‾‾‾‾‾‾√ the variance of pˆ is σ2pˆ= pˆ(1−pˆ)n

Which of the following is a way to estimate the success probability p associated with a Bernoulli trial?

Conduct n independent trials, and count the number of successes X. Estimate p with X/n.

Which of the following must be true about a Poisson process that produces events that occur in time?

Events cannot occur simultaneously. The number of events in disjoint intervals of time are independent.

True or False: The rate λ of a Poisson process can estimated as λˆ = Xt, where X is the observed number of events and t is the period of time or space where these events are observed.

False

Select the two quantities that are always equal in a Poisson distribution.

Mean Variance

Which of the following are denoted with parameter λ for a Poisson random variable X?

Mean Variance

The expression np is equal to which of the following?

Mean of a binomial random variable with parameters n and p.

Ten percent of the students at a large university are married. Students are polled one by one. Let X be the number of students polled up to and including the first married student. Select all that apply.

P(X = 25) =(0.924)(0.1). X ~ Geom(0.1).

Twelve students in a classroom of 20 students are females. Four students are chosen at random. Let X be the number of female students chosen. Which of the following statements is true?

P(X = 4) = (12/20)(11/19)(10/18)(9/17).

A random sample of 30 items was taken from a large shipment, and X = 2 items were found to be defective. Therefore the true proportion of defective items in the entire shipment is 2/30.

false

The Poisson distribution is an approximation to the binomial distribution when n is

n large and p is small

Sixty-five percent of the adults in a particular large city are married. A random sample of 7 adults is drawn from this city. Let X be the number of married people in this sample, and let p(x) be the probability mass function of X. Select all that apply.

p(x) = 7!x!(7-x)!0.65x 0.357 - x for x = 0,1,...,7 and 0 otherwise. X ~ Bin(7, 0.65).

If the sample proportion pˆ is used to estimate the probability of success in a Bernoulli trial, then _________.

pˆ is an unbiased estimate of p μpˆ = p

If we observe X events occurring over a period of t hours in a Poisson process, the mean rate per hour is estimated as λˆ = X/t. We determine the precision of the estimate by computing the _________ of λˆ.

uncertainty standard deviation

The number of cracks in a long sidewalk follows a Poisson process. A random sample of t square feet contained a total of 250 cracks. Let λ be the mean number of cracks per square foot. Estimate the value of λ.

λˆ = 250/t

The concentration of particles in a large batch of solution is unknown. The solution is thoroughly agitated and a sample of 0.25 mL is withdrawn. There are 12 particles in this sample. Estimate the concentration λ̂ in particles per mL, and find the uncertainty in the estimate.

λ̂ = 12/0.25 = 48. The uncertainty in λ̂ is σλ̂ = 480.25.

Let X be binomial with parameters n = 50 and p = 0.04, and let Y be Poisson with parameter λ = np = 2.

μX = μY = np. The distribution of X is approximately the same as the distribution of Y.