Stats Final

A friend offers to let you play a game in which you roll a fair six-sided die. If you roll a 6, your friend pays you $20. Otherwise, you pay $5. If you play this game one time, what is the variance of the amount of money you'll make?

$86.81

A sample of 50 students was taken from the local university. These students spent an average of $170 on books this semester, with a standard deviation of $25.50. Which of the following could you say with 95% confidence was the average spent on books by these 50 students?

(162.9, 177.1)

A friend offers to let you play a game in which you roll a fair six-sided die. If you roll a 6, your friend pays you $20. Otherwise, you pay $5. If you play this game one time, what is the expected amount of money you'll make?

-$0.833

Say a disease affects 60 in 100 people. A person with the disease tests positive 95% of the time, whereas a person without the disease tests positive only 10% of the time. If a randomly selected person tests positive, what is the probability that they have the disease?

.93444

Suppose that in a certain location, the diameter of dust particles in the air follow a log-normal distribution with parameters mu=2.5 and sigma = 1.8. If a particle is chosen at random, what is P(X > 1000)?

0.0071

A factory has three machines producing widgets. 20% of the widgets are produced by machine A, 30% of the widgets are produced by machine B, while the remaining 50% are produced by machine C. If you select 6 widgets at random, what is the probability that each machine produced exactly 2 widgets?

0.081

A product may exhibit 2 types of defects. Suppose the probability a product has no defect is 0.9, the probability that it has a type A defect (may also have type B) is 0.3, and the probability that the product has both type A and type B is 0.2. What is the probability the product has a type B defect?

0.09

A deck of 52 playing cards contains 12 face cards. You draw 3 cards at random. What is the probability that 2 or more are face cards?

0.12941

A friend offers to let you play a game in which you roll a fair six-sided die. If you roll a 6, your friend pays you $20. Otherwise, you pay $5. If you play this game $100 times, what is the approx. probability that you'll break even (make at least $0)?

0.1867

Suppose that the probability of event A is 0.2 and the probability of event B is 0.4. Also, suppose that the two events are independent. Then P(A|B) is:

0.2

What is the variance of the Bernoulli distribution with p=.6?

0.24

A product may exhibit 2 types of defects. Suppose the probability a product has no defect is 0.9, the probability that it has a type A defect (may also have type B) is 0.3, and the probability that the product has both type A and type B is 0.2. Given that a product has a defect, what is the probability that it has a type A defect?

0.3

Say a disease affects 60 in 100 people. A person with the disease tests positive 95% of the time, whereas a person without the disease tests positive only 10% of the time. What is the probability that a randomly selected person has the disease?

0.61

A study asked 940 Americans about their opinion on the death penalty. Among Americans with a bachelor's degree, 505 favor and 184 oppose. With only high school diploma, 159 favor and 92 oppose. Find a 95% confidence interval for the proportion of those who favor the death penalty among those who only have a high school diploma.

0.7329 += 0.330

The travel time for a businesswoman traveling between Dallas and Fort Worth is uniformly distributed between 40 and 90 minutes. The probability that she will finish her trip in 80 minutes or less is:

0.8

Suppose that a certain coin comes up heads with 70% probability. A one0tailed z procedure with alpha = 0.01 is used to test if the coin gives heads more than 50% of the time, based on 100 trials. What is the approximate probability that the test will correctly identify that the coin is unfair?

0.9656

Two friends agree to meet at 1:00 pm. The number of minutes late each friends arrived is independently normally distributed with mean -5 minutes (meaning they would be 5 minutes early) and a stdev of 10 minutes. Suppose the friend who arrives first waits 30 minutes after arriving before giving up and leaving. What is the probability that they will meet?

0.9660

A study asked 940 Americans about their opinion on the death penalty. Among Americans with a bachelor's degree, 505 favor and 184 oppose. With only high school diploma, 159 favor and 92 oppose. Find a 95% confidence interval for the difference between the two proportions.

0.995 +- 0.682

Two friends agree to meet at 1:00 pm. The number of minutes late each friends arrived is independently normally distributed with mean -5 minutes (meaning they would be 5 minutes early) and a stdev of 10 minutes. Suppose the friend who arrives first waits until 1:25 pm before giving up (and if neither arrives before 1:25 they both give up). What is the probability that the friends will meet?

0.9974

A distribution of data has a mean of 15 and a standard deviation of 2. How many standard deviations away from the mean is a value of 13? (give as number)

1

The Law of Large Numbers States that if you have n random variables that are independent , identically distributed (iid) random variables with mean 𝜇, then as 𝑛⟶∞, 𝑡ℎ𝑒 𝑠𝑎𝑚𝑝𝑙𝑒 𝑚𝑒𝑎𝑛 𝑋 converges to 𝜇, with what probability.

1

The probability mass function has to always sum to?

1

What Z-value is associated with a 90% confidence interval?

1.65

What Z-value is associated with a 95% confidence interval?

1.96

A system contains two components, and both must work for the system to work. A->B The lifetime of each component follows an exponential distribution with unknown lambda. Assume they fall independently of one another. If we observe that the system fails after 152 days, what is the maximum likelihood estimate for the parameter lambda?

1/304

Suppose we roll two fair six-sided die. What is the probability that they add up to 3 or 11?

1/9

For the exponential probability distribution, what is the mean?

1/lambda

For the exponential probability distribution, what is the standard deviation?

1/lambda

A random sample of 72 statistics students was taken to estimate the proportion of students who also were in the Math Club. The 90% confidence interval was 0.438 to 0.642. Using this information, what size sample would be necessary to estimate the true proportion to within a margin of error of 0.08 using 95% confidence?

195

What Z-value is associated with a 99% confidence interval?

2.58

You are expecting a call from two friends. Suppose time X until Alice calls is a random variable with pdf fx(x) = x/200 on [0, 20] while the time until Bob calls is a random variable with pdf fy(y) = y/450 on [0,30], independent of the time Alice calls. What is P(Bob calls before Alice)?

2/9

How many ways are there to rearrange the letters in the word bookkeeper?

21772800

Two different alloys are being considered for making lead-free solder used in the wave soldering process for printed circuit boards. A crucial characteristic of solder is its melting point, which is known to follow a Normal distribution. A study was conducted using a random sample of 21 pieces of solder made from each of the two alloys. In each sample, the temperature at which each of the 21 pieces melted was determined. The mean and standard deviation of the sample for Alloy 1 the results were m1 = 218.9ºC and s1 = 2.7ºC; for Alloy 2 the results were m2 = 215.5ºC and 𝑠2= 3.6ºC. If we were to test 𝐻0 : 𝜇1=𝜇2 against 𝐻𝐴: 𝜇1≠𝜇2 , what would be the value of the test statistic?

3.46

You randomly choose two marbles from a jar. The jar contains 3 red marbles, 10 blue marbles, 8 greenmarbles, and 4 yellow marbles. What is the probability of choosing two blue marbles?

3/20

Given the following pdf 𝑓(𝑥)=𝑘𝑥^2 for 0≤𝑥≤2, 0 𝑜𝑡ℎ𝑒𝑟𝑤𝑖𝑠𝑒, Find the value of k.

3/8

A system contains two components, and both must work for the system to work. A->B The lifetime of each component follows an exponential distribution with unknown lambda. Assume they fall independently of one another. If we observe that the system fails after 152 days, what is the max likelihood estimate for the mean lifetime of a component?

304

Let X and Y be random variables and let c be a constant. if the V(X) = 3.5 and c =3 Then V(cX) = ?

31.5

Consider a class survey of 259 students in which students report an average of 350 minutes on a typical weeknight. The sample stdev is 67 minutes. Find a 99% confidence interval for the mean amount of time the students would report studying on a typical weeknight.

350 +- 10.7

Suppose we roll two fair six-sided dice. What is the probability that they add up to 8?

5/36

If the population distribution of a variable is (approximately) normal, roughly ___% of the values should be above the mean.

50

If the population distribution of a variable is (approximately) normal, roughly ___% of the values should be below the mean.

50

How many different license plates can be made with two letters followed by three numbers? Hint: 26 letters in the alphabet and 10 possible digits

676,000

Let X and Y be random variables and let c be a constant. if the E(X) = 1.5 and c = 5 Then E(cX) =

7.5

In testing a hypothesis about two population means, it the t distribution is used, which of the following assumptions is required?

Both populations are normally distributed.

In uniform distribution, length of a rectangle is the ___________ between a and b.

Difference

What statistical distribution is used for testing the difference between two population variances?

F

Are two random variables independent when one variable is normal and the other is exponential?

No

Does the cdf range from -1 to 1?

No

For the normal probability distribution, is the mean and standard deviation the same?

No

If the p-value is 0.039 and alpha is 0.01, should H_0 be rejected?

No

If the p-value is 0.84 and alpha is 0.05, should H_0 be rejected?

No

If two random variables are uncorrelated, is p = -1?

No

Is H: median = 5 a legitimate statistical test?

No

Is V(X) = E[X^2] - 𝜎^2?

No

Is the Z value also called the student value?

No

Is the Z value often called the error value?

No

If three events A,B,C are independent means that the P(A ∩ B ∩ C) = ?

P(A)P(B)P(C)

Two random samples have sizes of n=49 and n=36 respectively. Is the confidence interval for n=49 narrower or wider than n=36?

Wider

A study asked 940 Americans about their opinion on the death penalty. Among Americans with a bachelor's degree, 505 favor and 184 oppose. With only high school diploma, 159 favor and 92 oppose. Does the data provide sufficient evidence to conclude that the proportion who favor the death penalty is the same regardless of education level? Let alpha = 0.01

Yes

Are two random variables independent when their joint pdf = the product of their marginal pdfs?

Yes

As the sample size grows, does a a t-distribution's gradually approach the normal distribution?

Yes

Consider a class survey of 259 students in which students report an average of 350 minutes on a typical weeknight. The sample stdev is 67 minutes. If alpha = 0.01, does the study give good evidence that students report studying less than 6 hours per weeknight on average?

Yes

Does the cdf range from 0 to 1?

Yes

For the exponential probability distribution, is the mean and standard deviation the same?

Yes

If A ⊆ B and B ⊆ C, then is A contained in C?

Yes

If X and Y are independent normal random variables, then are their sum X + Y also a normal random variable?

Yes

If the p-value is 0.0498 and alpha is 0.05, should H_0 be rejected?

Yes

If the population distribution of a variable is (approximately) normal, should the mean and median be the same?

Yes

If two random variables are uncorrelated, is p = 0?

Yes

Is H: mean = 0 a legitimate statistical test?

Yes

Is H: p = 0.5 a legitimate statistical test?

Yes

Is H: sigma >= 100 a legitimate statistical test?

Yes

Is V(X) = E[X^2] - E[X]^2?

Yes

Is V(X) = E[X^2] - 𝜇^2?

Yes

Is a t-distribution symmetric?

Yes

Is a t-distribution's exact shape (spread) characterized by the degrees of freedom?

Yes

Is the Z value also called the critical value?

Yes

Standard deviation of a uniform distribution is given by___________.

b-a/sqrt(12)

Is the exponential probability distribution continuous or discrete?

continuous

If you test for the for the mean of the differences in a paired t-test, there are how many degrees of freedom?

n-1

What is the variance of the binomial distribution?

np(1-p)

Suppose we were interested in determining if there were differences in the average prices among twolocal supermarkets. We randomly pick the same six items to compare at both supermarkets. Which statisticalprocedure would be best to use for this study?

paired t-test

If X and Y are independent normal random variables, then what is the expected value of their sum X + Y ? Given that X is normal with mean 𝜇1 and standard deviation 𝜎1 and Y is normal with mean 𝜇2 and standard deviation 𝜎2.

𝜇1 + 𝜇2