Stats Test 3

c, d

.5 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_______ Check all that apply a) X~Bin(40,0.05) b) for x=1,2,..,40, p(x) = \frac{40!}{x!(40-x)!}0.05^x 0.95^(40-x),p(x) = 0 otherwise c) the variance of the number of defective items per box is 40(0.05)(0.95) d) the mean number of defective items per box is \mu_x = 40(0.05)

a - 1,3; b - 2

6.Match each of the following to the appropriate formula for a Poisson random variable X with parameter λ Drag each item into the appropriate category. Categories: λ (a) ,√{λ} (b) 1) mean 2) standard deviation 3) variance

2

A Bernoulli trial requires that when performing an experiment,the experiment can be viewed as having only _____ possible outcomes

b,c

A biased coin has probability 0.2 landing tails. Each student in your class tosses this coin 5 times and records as X the number of tails in the 5 tosses. Select all that apply. a) The possible values of X(number of tails each student may get)are 1,2,3,4,5. b) The mean number of tails for each student is \mu = 1 c) X has a binomial distribution with parameters n=5 and p=0.2,i.e.,X~Bin(5,0.2) d) The standard deviation of X is 5(0.2)(0.8)

Ans:c-a-b-d

A biased coin with probability 0.7 of landing tails is tossed 10 times. Then another biased coin with probability 0.2 of landing tails is tossed 5 times. The probability that the total number of tails in the 15 tosses is greater than 8 can be estimated using simulation by the following steps: Place these in the proper order. a) Compute the number of 1's in the 15 random numbers: T =\sum_{i=1}^{10} X_i + \sum_{j=1}^5 Y_j b) Repeat step 1 and step 2 1000 times each. c) Generate 10 random numbers X1,..,X_{10} from Bernoulli(0.7) and 5 random numbers Y_1,..,Y_5 from Bernoulli(0.2) d) Obtain the proportion of times out of 1000 that T is greater than 8

b,c,d

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: a) If the coin is tossed 4 times, each arrangement of 1 head in 4 tosses has probability 0.25^3(0.75) b) The number of arrangements of 1 head in 4 tosses is 4!/1!(4-1)! c) P(x=2) = (number of arrangements of 2 heads in 4 tosses)(0.25^2)(0.75^2) d) The possible arrangements of 1 head in 4 tosses are{HTTT,THTT,TTHT,TTTH}

a, b

A coin with probability 0.7 coming up tails is tossed. Let X =1 if a head comes, and X = 0 otherwise. Check all that apply a) X~Bernoulli(0.3) b) The variance of X is \sigma_x^2 = (0.3)(0.7). c) The mean of X is \mu_x = 0.7 d) X~Bernoulli(0.7)

Ans: b,c

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. Check all that apply a) The estimator of p is \hat{p} = 41/959 b) The estimator of p is \hat{p} = 41/1000 c) The uncertainty in \hat{p} is \sqrt{(0.04)(0.959)/1000} d) The uncertainty in \hat{p} is (0.04)(0.959)/1000

0.16/2

A digital multimeter is used to measure the voltage of a used AA battery. Four independent measurements were taken. The sample mean was 1.12 volts with standard deviation s= 0.16 volts. What is the uncertainty in the sample mean? 4(0.16) 2(0.16) 0.16/2 0.16 0.16/4

d-a-c-b

A manufacturing process produces squared items whose lengths Lin inches are normally distributed with mean \mu =5 and standard deviation \sigma = 0.02. The standard deviation \sigma_A of the area of the squares can be estimated using simulations by conducting the following steps: Place these in the proper order a) Compute the square of the randomly generated numbers: A_i =(L_i)^2, i=1,2,..,1000 b) Obtain the sample standard deviation of the 1000 simulated areas A_i:S_A c) Repeat step 1 and step 2 1000 times d) Generate 1000 random numbers L_i, i=1,2,..,1000 from a normal distribution with mean \mu = 5 and standard deviation \sigma = 0.02

generator

A random number ________ is a procedure that produces values that have the same statistical properties as a random sample from specified distribution.

-The bias is 1 ppb and the uncertainty is 3/ √(5) ppb

A sample of gas is known to contain benzene in a concentration of 100 ppb. The concentration is measured five times. The five measurements average 101 ppb with standard deviation s = 3 ppb. Which statement best describes this measuring process? -Neither the bias nor the uncertainty can be estimated. -The bias is 1 ppb and the uncertainty cannot be estimated. -The bias is 1 ppb and the uncertainty is 3/ √(5) ppb -The bias cannot be estimated and the uncertainty is 3/ √(5) ppb.

Ans: b-d-a-c

A system consists of three components A,B, and C connected in parallel. The system will continue functioning until all 3 components fail. Assume that the lifetime of these components are independent and exponentially distributed with parameter \lambda=3. The following steps can be used to estimate the mean lifetime of the system. Place these in the proper order a) Repeat steps 1 and 2 to stimulate 1000 lifetimes L_i = max(A_i,B_i, C_i), i=1,2,..,1000, of the entire system. b) Generate three random numbers A_i,B_i,C_i from an exponential distribution with \lambda = 3 to simulate the lifetime of three components. c) Estimate the average lifetime of the system by computing the sample average L of the 1000 simulated lifetimes. d) Obtain the maximum of 3 random numbers A_1, B_1, C_1: L_1 =max(A_1, B_1, C_1) to simulate one lifetime of the sytyem.

d

Assume that heights of American women aged 18 to 24 are normally distributed with mean µ=166 cm and standard deviation σ = 6.3cm. Let z be the value for which the area to its right under the standard normal curve is 0.20. What is the 80th percentile of the heights a) 6.3 + z(166) b) 6.3 - z(166) c) 166 - z(6.3) d) 166 + z(6.3)

bias, random

Error in a measurement can be thought of as being composed of two parts: systematic error, also called ______, and ________ error

Ans: a , b

Five percent of all items manufactured by a company are defective. These items are packaged in boxes of 10. One of there boxes is randomly sampled, and X is the number of defective items in the box. Select all that apply. a) X~Bin(10, 0.05) b) The mean number of defective items in a box is 10(0.05) c) The standard deviation of the number of defective items in a box is 10(0.05)(0.95) d) The probability mass function of X is p(x) = 0.05^x 0.95^(10-x)for x =0,1,..,10 and p(x) = 0 otherwise.

b, c

For any Bernoulli trial, a Bernoulli random variable X can be defined. Which of the following are true statements about X? Check all that apply a) If the experiment results in success, X = 0. Otherwise, X =1. b) If the experiment results in success, X = 1. Otherwise X = 0. c) 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=1) = p. d) The probability mass function of X is given by p(0) = p(X =0) = p, where p is the probability of success, and p(1) = p(X=1) = 1 - p.

standard

For data from a normal population, z-scores are from the ____ normal population.

-coefficient of variation -relative uncertainty

If an uncertainty is represented as a fraction of the true value, it is called the -coefficient of variation -absolute uncertainty -relative uncertainty -relative variation

tails

If the normal distribution is used to approximate the Poisson distribution for areas in the _________ of the curve, the continuity correction sometimes makes the approximation worse.

0; 1

If x is a value sampled from a normal distribution with mean \µ and standard deviation \σ , the z-score of x is an item sampled from a normal distribution with mean ___ and standard deviation ____.

c-a-b-d

If x_1,..,x_7 is a random sample of size 7 from a uniform distribution with parameter A=0, B =1, then the simulation can be used to determine whether the sample mean X = (1/7) \sum x_i is normally distributed by using the following steps. Place these is the proper order a) Repeat step 1 1000 times to obtain 1000 sample mean \bar{X}_i,i=1,..,1000 b) Construct a histogram or a normal probability plot of the 1000 sample means X_i, i=1,..,1000 c) Generate 7 random numbers x_1,..,x_7 from a uniform distribution with parameters A=0 and B=1 and compute their mean \bar{X}. d) Determine if the histogram has a bell shape or if the points in the normal probability plot are on a straight line.

1-b, 2-d, 3-a, 4-c

Let X be a binomial random variable, and let \a and \b be a constant with a<b. The normal approximation will be used to compute binomial probabilities. Match each probability to the appropriate area under the normal curve. 1) P(a <= X <= b) 2) P(X >= a) 3) P(X <= b) 4) P(X = a) a) area to the left of b+0.5 b) area between a-0.5 and b+0.5 c) area between a-0.5 and a+0.5 d) area to the right of a-0.5

Ans: a-d-c-b

Let X_1, X_2,..,X_10 be a random sample from a normal population N(3,25). The bias in the sample standard deviation \s for the population standard deviation \σ can be estimated using simulation as follows: a) Generate 10 random numbers X_1,X_2,..,X_10 from a normal distribution N(3,25) b) Estimate the bias in estimating \sigma with s as \hat{S} -5 c) Obtain the average \hat{S} of the 1000 simulated s_i's d) Repeat 1000 times to simulate 1000 sample standard deviation s_i, i = 1,2...,1000 of 10 random numbers from N(3,25)

c

Let X_1,X_2,...,X_n be a simple random sample from a population. Let \bar{X} be the sample mean and S be the sum. Which of the following conditions must be met for \bar{X} and S to be approximately normally distributed? a) X_1,X_2,..,X_n must be normally distributed b) The sample size must be less than 30 c) The sample size mast be greater than 30

Ans: a,c

Let X_1=-1, X_2=0 and X_3=1 be a random sample from a normal population with mean \mu and variance \sigma^2, i.e., N(\mu,\sigma^2). Select that all apply. a) The mean \mu can be estimated as \bar{X}=0 b) The variance \sigma^2 can be estimated as s^2 = \frac{(-1-0)^2 +(0-0)^2 + (1-0)^2}{3} c) The uncertainty in estimating \mu with \bar{X} can beapproximated by \frac{s}{\sqrt{n}}

Ans: b

Let X~Bin(n,p). Under what conditions can we use the Central Limit Theorem to approximate the distribution of X? a) n > 30 b) np > 10 and n(1-p) > 10 c) np > 30 and n(1-p) > 30 d) np >30

c

Let X~Bin(n,p). Under what conditions we can use the Central Limit Theorem to approximate the distribution of X? a) n> 30 b) np > 30 c) np > 10 and n(1-p)>10 d) np >30 and n(1-p)>30

a,c

Let X~N(12,25), and let Y=3X+7. Select all that apply. a) Y is also a normal random variable. b) The mean of Y is 3(25) +7 c) The variance of Y is 3^2(25) d) The standard deviation of Y is 3(25)

a,c

Let X~N(\mu, \sigma^2), and let Y = aX+b, where a and b are constants with a not equal to zero. Select all that apply. a) the mean of Y is a\mu + b b) The variance of Y is a a^2\sigma^2 +b c) Y is also normal random variable d) The standard deviation of Y is a\sigma +b

c,d

Let x be a random variable whose distribution is N(\mu,\sigma^2). Select all that apply a) the mean of the random variable X is \mu b) if x_1,..,x_n is a random sample from n(\mu,\sigma^2), then \sigma^2 can be estimated as s = \sqrt{1/(n-1) s\\sum_{i=1}^{n} (x_i- \hat{X})^2} c) if x_1,..,x_n is a random sample from N(\mu,\sigma^2), then \mu can be estimated as \hat{x} = 1/n \sum_{i=1}^{n} X_i d) The variance of the random variable X is \sigma^2

Ans: a,b,c

Minimum daily temperatures in degrees Fahrenheit during the winter in a city in New England are normally distributed with a mean temperature of 27F and a standard deviation of 5F. Let F be the temperature in degrees Fahrenheit on a certain day, and let C =(5/9)(F-32) be the temperature in degrees Celsius. Select all that apply a) The variance of the temperatures in degrees Celsius is \sigma_y^2= (5/9)^2(5)^2 b) The random variable C = (5/9)(F-32) also has a normal distribution c) C = (5/9)(F-32) is a linear function of the random variable F d) The mean temperature in degrees Celsius is \µ_c = (5/9)(32)

a,b,d

Of the customers who purchase gas at a certain station, 45 percent purchase regular gas, 35 percent purchase plus, and 20 percent purchase supreme. Assume no customer buy more than 1 type of gas. For the next customer who purchases gas at this station, let X=1 if the customer purchases regular gas and X=0 otherwise. Also,let Y =1 if the customer purchase plus gas and Y=0 otherwise. Select that apply. a) Y~Bernoulli(0.35) b) X+Y=Z where Z~Bernoulli(0.80) c) It is possible for the product XY to be equal to 1 d) X~Bernoulli(0.45)

c-a-b-d

Put in order the following steps required to construct a normal probability plot for a random sample x_1,..,x_n from a population Place these in the proper order. a) Assign increasing evenly spaced values between 0 to 1 to each X_i using the formula (i-0.5)/n b) Compute the normal quartiles Q_i = X + z_(i-0.5)/n is the(i-0.5)/n th quartile of N(0,1) c) Order the sample elements X_1,..,X_n from smallest to thelargest, and compute the sample mean X and the sample standarddeviation s. d) Plot values(X_i, Q_i)

-The uncertainty is small.

Repeated measurements of a quantity. are made. The values are all very close to each other. What do we know about this measuring process? -The uncertainty is large. -The bias is large. -The uncertainty is small. -The bias is small.

normally

The Central Limit Theorem states that for sufficient large n (in general n >30) the distribution of the mean \bar{X}_n and the sum S_n of a sample of size n are approximately _____________ distributed regardless of the distribution of the population from which these samples are drawn.

mean

The bias in a measuring process is the difference between the __________ measurement and the true value.

-The bias cannot be estimated, and the uncertainty is 2/ √(6) degrees.

The boiling point of a chemical is measured six times. The measurements average 65 degrees Celsius, with standard deviation s = 2 degrees Celsius. Which statement best describes this measuring process? -The bias is 2 degrees, and the uncertainty is 2/ √(6) degrees. -The bias cannot be estimated, and the uncertainty is 2/ √(6) degrees. -Neither the bias nor the uncertainty can be estimated. -The bias is 2 degrees, and the uncertainty cannot be estimated.

-Accurate but not precise

The boiling point of water at sea level was measured four times. The results, in degrees Celsius, were 97.1, 102.2, 101.6, and 99.1. Which of the following statements best describes this measuring process? -Accurate but not precise -Both accurate and precise -Neither accurate nor precise -Precise but not accurate

-Precise but not accurate

The freezing point of water was measured five times. The results, in degrees Celsius, were 5.41, 5.42, 5.44, 5.41, and 5.43. Which of the following statements best describes this measuring process. -Neither accurate nor precise -Precise but not accurate -Accurate but not precise -Both accurate and precise

b,d

The number of trees of a particular species in a large wooded area follows a Poisson distribution with mean \lambda trees per acre where \lambda is unknown. In 50 randomly selected one-acre plots of its wooded area, you count a total 750 trees of this species. Estimate \lambda and find the uncertainty in the estimate. Check all that apply. a) the uncertainty in \hat{\lambda} is \sigma = \sqrt{750/50} b) the uncertainty in \hat{\lambda} is \sigma = \sqrt{15/50} c) The estimate of \lambda is \hat{\lamda} = 750 d) The estimate of \lambda is \hat{\lamda} = 750/50=15

0.3

The perimeter of an equilateral triangle is measured to be 12.6 ± 0.9 cm. The length of a side will be estimated as 4.2 ± _____ cm.

large; small

The poisson distribution is an approximation to the binomial distribution when n is __ and p is __

Ans: d

The quantity \sqrt{np(1-p)} is the ___ of a binomial random variable with parameters n and p a) probability b) expected value c) mean d) standard deviation e) variance

0.4

The side of a square is measured to be 4.2 ± 0.1 cm. The perimeter of the square will be estimated as 16.8 ± ____ cm.

accurate, precise

The smaller the bias, the more ______ the measuring process, and the smaller the uncertainty, the more _____ the measuring process

-The temperature was measured to be 18.2 -The uncertainty in the measurement is 0.3

The temperature of a solution is given as 18.2 ± 0.3 degrees Celsius. Which of the following statements are true? -The temperature was measured to be 18.2. -The bias in the measurement is 0.3. -The true temperature is 18.2. -The uncertainty in the measurement is 0.3.

1.5/3 ksi

The uncertainty of a device used to measure the compression strength of cans is 1.5 ksi. If the compression strength is measured for a sample of 9 cans from a particular manufacturer, what is the uncertainty in the average of these measurements? - 1.5/9 ksi - 1.5 ksi - 3(1.5) ksi - 1.5/3 ksi - 9(1.5) ksi

0.3/7.1

The volume of a cylinder is measured to be V = 7.1 ± 0.3 cm^3. Estimate the relative uncertainty in V. - ln 0.3 - 0.3/7.1 - 0.3 - 7.1/0.3

-The uncertainty in the measurement is 0.2. -The weight was measured to be 2.3.

The weight of a package is given as 2.3 ± 0.2 kg. Which of the following statements are true? -The bias in the measurement is 0.2. -The true weight is 2.3. -The uncertainty in the measurement is 0.2. -The weight was measured to be 2.3.

a

Thirty percent of students at a large university are married. A sample of 100 students is drawn. Let X be the number of married students in the sample. To use the continuity correction to compute the probability that 30 <= X <= 40 which probability do we actually compute? a) P(29.5<X<39.5) b) P(30.5<X<40.5) c) P(30.5<X<39.5) d) P(29.5<X<40.5)

-True

True or false: Estimates of uncertainty based on small samples are often crude. -True -False

Ans:d

Twelve students in a classroom of 20 students are females. Four student are chosen at random. Let X be the number of female students chosen. Which of the following statements is true? a) X has a binomial distribution with b = 4 and p = 0.6 b) p(x=4) = 0.6^4 c) There are 4 Bernoulli trials in this experiment that can be d) p(x=4) = (12/20)(11/19)(10/18)(7/7)

A, B , C

Two adjacent sides of a rectangular lot are measured. The measurements are X and Y. These measurements are independent, and their uncertainties are σx and σy, respectively. Which of the following statements are true?

Random Error

What kind of error changes from measurement to measurement? -Systematic Error -Random Error -Bias

Ans:b

What z-score corresponds to the 20th percentile of the normal curve? a) z = -2.88 b) z = -0.84 c) z = 0.84 d) z = 2.88

-The second derivative d^2U/dX^2 is small. -The uncertainty σx of X is small.

When a measurement X is unbiased, the bias in a nonlinear function of X, U= U(X), will generally be small so long as which of the following conditions hold? -The second derivative d^2U/dX^2 is small. -The uncertainty σx of X is small. -The first derivative dU/dX is small.

b

When making beer, 2493 yeast cells are counted in 0.1 mL of a suspension. Assume that the number of yeast cells follows a Poisson process, and let λ be the number of cells per mL. Estimate the value of λ a) \lamda = 2493(0.1) b) \lamda = 2493/0.1 c) \lamda = 2493^0.1

bias, uncertainty

When the true value of a quantity is unknown, we cannot estimate the _________ from repeated measurements but we can estimate the _____________.

-They are only rough approximations to the true uncertainty. -They should be expressed with no more than two significant digits.

Which of the following are true about propagation of error uncertainties? -They are only rough approximations to the true uncertainty. -They are equal to the true uncertainty. -They should be expressed with no more than two significant digits. -They should be expressed with the same number of significant digits as the measurement.

-Uncertainty represents precision. -Bias represents accuracy

Which of the following are true about the bias and uncertainty of a measurement? -Uncertainty represents precision. -Bias represents precision. -Uncertainty represents accuracy. -Bias represents accuracy

a, b

Which of the following conclusions can be made from this normal probability plot? (purple line and dots-line) Check all that apply a) The data come from a population that is not close to normal b) The points are not, in general, close to a straight line c) The data come from a population that is close to normal d) The points are, in general, close to the straight line

Ans: c

Which of the following is a way to estimate the success probability p associated with a Bernoulli trial? a) Estimate the probability of success as \hat{p} = X, where X is the number of success in the n trials. b) Conduct a Bernoulli trial, and estimate p=1 if the trial is success and p=0 if the trial is a failure. c) Conduct n independent trials, and count the number of successes X. Estimate p with X/n.

The standard deviation σ

Which of the following represents the uncertainty in a measuring process? - The sample mean x̄ -The mean µ - The standard deviation σ

b, c

Which of the following statements are true regarding how to interpret a probability plot? Check all that apply a) A single point in the plot that is very far from the line should not be a matter of attention b) If the points of the plot lie close to a straight line, it is reasonable to believe that the population is close to normal c) Points in the plot corresponding to the points at the ends (high or low) may be a bit far from the line, even when the population is close to normal d) As a rule of thumb, probability (or Q_Q) plots are more reliable when there are fewer than 30 points

Ans:b

Which z-scores have 90% of the area under the normal curve % between them? a) z_1 = -2.65 and z_2 = 2.65 b) z_1 = -1.645 and z_2 = 1.645 c) z_1 = -1.28 and z_2 = 1.28 d) z_1 = -1.96 and z_2 = 1.96

A

Wood pegs are to be tightly fitted into a wood board. The diameters of the pegs are 1.5 ± 0.01 cm. The drill that is used to bore the holes in the board has diameter 1.55 ± 0.02 cm. The clearance is the difference between the diameter of the hole and the diameter of the peg. The clearance is estimated as 1.55 - 1.50 = 0.05. The uncertainty in this estimate is

Simulation

__________ refers to the process of generating random numbers and treating them as if they were data generated by an actual scientific experiment.

a, b

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. a) The probability that there are exactly 8 particles in the sample is exp{-15}15^8/8! b) µ_x = 15 c) X~Poisson(5) d) The probability that there are at least three particles in the sample is \exp{-15}15^3/3! e) \sigma_x^2 = \sqrt{15}