Show:
You linearly regress
I prefer to eat on two-way tables
That's so Random, Variable.
Shampoo and Conditioning
Mixed Lunch
100
Three things we should be cautious of when doing linear regression.
What are (for example), extrapolation (we should not use linear regression to do this), lurking variables, confounding variables, outliers, influential values...
100
In a 4 by 7 two-way table, this is the number of possible values the response variable has.
What is 4? (Recall the response variable is listed in the rows of the table)
100
This is the expected value of roll of a 4-sided die which is weighted so that 1 and 2 are rolled three times as often as 3 and 4.
What is 2? Call p = P(3) = P(4), so that P(1) = P(2) = 3p. Observe that we must have 3p + 3p + p + p = 1, so that p must equal 1/8. Then E(X) = 1(3/8)+2(3/8)+3(1/8)+4(1/8) = 16/8 = 2.
100
A number is chosen uniformly at random from the interval (0,3). What is the probability the number is between 1.2 and 2.2 given that it is between 1.5 and 2.5?
What is 0.7?
100
The mean of a density curve can be thought of as this.
What is the balancing point of the curve?
200
This is how you find a linear regression line equation.
What is m=(r)(s_y)/(s_x), b=(y_bar) - m(x_bar), where r is the correlation coefficient, s_y is the standard deviation (sd) in the response variable, s_x is the sd of the explanatory variable, y_bar is the mean of the response variable, and x_bar is the mean of the explanatory variable. Then m gives the slope, b gives the y-intercept and the equation of the line is y=mx+b.
200
These are the marginal distributions for the two-way table Red Blue Other Men 20 30 12 Wom 25 35 10
What is Red Blue Other TOTAL Men 20 30 12 62 Wom 25 35 10 70 TOTAL 45 75 22 142
200
A variable X is normally distributed with mean 3 and variance 4. This is the mean and variance of the variable Y = 2X - 1.
What is 5 and 16?
200
Suppose a girl at birth will weigh less than 8lbs with probability 0.8 and a boy will weigh less than 8lbs with probability 0.6. If Sheila has a baby, what is the probability her baby will weigh less than 8lbs?
What is 0.7? Let X be the even the baby weighs less than 8lbs, and B/G the events that the baby is a boy/girl. Then by the Law of Total Probability, P(X) = P(X|B)P(B) + P(X|G)P(G) = (0.6)(0.5)+(0.8)(0.5) = 0.7.
200
This is the derivation of the rule that P(Ac|B) = 1 - P(A|B).
What is: 1 - P(A|B) = 1 - P(A and B) / P(B) = P(B)/P(B) - P(A and B)/P(B) = [ P(B) - P(A and B) ] / P(B) = P(Ac and B) / P(B) = P(Ac|B). [There are many other ways to show this.]
300
The word we will never use when using correlation.
What is "causes." Correlation does not imply causation!
300
What is the joint distribution for the two-way table Red Blue Other Men 20 30 12 Wom 25 35 10
What is Red Blue Other TOTAL Men .14 .21 .08 .43 Wom .18 .25 .15 .58 TOTAL .32 .46 .23 1
300
If the variance of X is 1 and the variance of Y is 1, this is the variance of X+Y.
There is not enough information! If the random variables X and Y are independent, the variance of X+Y is 1+1=2. Otherwise, the correlation between them must be known.(See Section 4.4)
300
Suppose a girl at birth will weigh less than 8lbs with probability 0.8 and a boy will weigh less than 8lbs with probability 0.6. If Sheila has a baby that weighs less than 8lbs at birth, what's the probability it is a girl?
What is ? P(G|X) = P(X|G)P(G)/P(X) = (0.8)(0.5)/(0.7) = 0.57. (See also the $200 question)
300
This is the Law of Large Numbers.
What is: When observing n independent trials each with expected value mu and taking the average of the outcomes, the sample average tends toward the true mean mu as n grows large. (See Section 4.4)
400
This is the residual of a data value.
What is the error between the predicted value and the true value (ie. the distance between the point on the scatterplot and the line)
400
This is the conditional distribution for men for the two-way table Red Blue Other Men 20 30 12 Wom 25 35 10
What is Red Blue Other .32 .48 .19
400
This is the variance of the outcome of a 4-sided die roll.
What is 1.25? Since E(X) = 1(1/4)+2(1/4)+3(1/4)+4(1/4) = 2.5 and E(X^2) = 1(1/4)+4(1/4)+9(1/4)+16(1/4) = 7.5, we have Var(X) = 7.5 - 2.5^2 = 1.25.
400
A certain IQ test is such that a "silly" person passes the test 90% of the time and an "unsilly" person passes it 12% of the time. Suppose that 50% of people are considered "silly." This it the probability that a person who failed the test is "unsilly."
P(pass test) = 0.9*0.5 + .12*0.5 = 0.51 by Law of Total Probability. Then by Bayes, P(unsilly | fail) = P(fail | unsilly)P(unsilly)/P(fail) = (0.88)(0.5)/0.49 = 0.90.
400
The following is true or false: When rolling two dice, the event A that the first die is an odd number and the event B that the sum of the dice is 8 are independent events.
Check: P(A) = 1/2, P(B) = 5/36, and P(A and B) = 2/36. Since P(A and B) is not equal to P(A)P(B), the events are not independent.
500
These are the units of the correlation coefficient r.
What are none? (No units)
500
This is the type of experimental design that reduces bias in the experimenter and allows for a representative sample from the population.
What is randomized block design?
500
A game is played in which a fair coin is tossed until a Heads appeared. The player's winnings starts at $1 and doubles each time a Tails is flipped. This is the expected value of the player's winnings.
What is INFINITY?! Let X be the winnings. Notice that he wins $1 if the coin lands H the first time (H), $2 if the coin lands T then H (TH), $4 if it lands TTH, $8 if it lands TTTH, and so on. Thus E(X) = 1(1/2) + 2(1/4) + 4(1/8) + 8(1/16) + .... = 1/2 + 1/2 + 1/2 + 1/2 + ....
500
There are four coins in a bag. One coin is a fair coin, one has "Tails" on both sides, and two have "Heads" on both sides. If you reach into the bag and draw a coin and flip it, what's the probability it lands Heads?
What is 0.625? Using the Law of Total Probability, P(H) = P(H | fair coin)P(fair coin) + P(H | heads coin)P(heads coin) + P(H | tails coin)P(tails coin) = (0.5)(0.25) + 1(0.5) + 0(0.25) = 0.625.
500
Joe buys a ticket in the Lottery every day, always betting on 456. He will win something if the winning three digit number (from 000 through 999) contains a 4, 5, and 6 (in any order). Each day he wins or not independently of the other days. What is the probability Joe will win for the first time on the tenth day?
Notice that on a given day, the probability that he wins is 6/1000 = 0.006, and that he loses is 1 - 0.006 = 0.994. The probability that he wins for the first time on the tenth day is the probability that he loses the first nine days and wins the tenth day. By independence, this is equal to (0.994^9)(0.006) = 0.0057.