1.State whether the following statements are true or false, and provide a brief
i. For a set of observations x1, x2,...,xn,
set of observations x1, x2,...,xn, with mean ¯x, then:
(xi x¯) > 0.
ii. For two independent events A and B such that P(A) > 0 and P(B) > 0,
P(A [ B) < P(A) + P(B).
iii. For a random variable X, E(X2) can be less than (E(X))2.
iv. Rejecting a true null hypothesis is known as the power of a test.
v. A 4-by-2 contingency table which results in a 2 test statistic value of
6.724 is statistically significant at the 5% significance level.
2.11. In a game, you draw thirteen cards with replacement from a deck of playing cards. If you draw any
y aces or twos, you lose the game immediately. You also lose if you draw picture cards(J,Q,K) more than twice. In this question, you’ll study the probability of winning this game.(a) What is the probability of drawing no aces or twos after thirteen draws?(b) Given you have drawn thirteen times, none of which is aces or twos, what is the probability that you draw at most two picture cards?(c) What is the probability to win this game?
12. Suppose you are tossing an unbiased coin for100times.(a) What is the probability of getting50heads and50tails?(b) LetXbe the random variable counting the number of heads you observe in this exper-iment. What is the expected value ofX? What is the variance ofX? What is thestandard deviation ofX?
13. The following are probability distributions for two random variablesX,Y.
(a) Construct the probability distribution table for the random variableXY.(b) Find E[X],E[Y] and E[XY]. Is is true that E[XY] =E[X]E[Y]?(c) Find the variances σ2X,σ2Y,σ2XY of X,Y and XY. Is it true that σ2XY=σ2Xσ2Y?
14. The aliens who are fond of gambling came back to play another game with you. In this game, you first toss a coin5times. If you observe3or fewer tails, you roll a die3times. If youobserve4or more tails, you roll a die20times. What is the probability that you end up with at most two6’s in your dice rolls?
15. (Challenge question, worth2points) You have two bags, each of which contains10marbles.Each time you remove a marble from a random bag. What is the probability that after one of the bags is emptied, there are still exactly3marbles in the other bag?
4.Credit card sales The National Association of Retailers reports that 62% of all purchases are now made by credit card;
de by credit card; you think this is true at your store as well. On a typical day you make 20 sales.
Show that this situation can be modeled by a binomial distribution. For credit, you must discuss each of the criteria required for a binomial experiment.
Define the random variable x in this scenario, using the context of the problem.
List all possible values of x for this situation.
On one trial for this scenario, what does “success” mean? Explain using the words of the problem.
What is the probability of success in this scenario?
What is the probability of failure in this scenario?
Probability Distribution Instructions¬¬¬¬
In Excel, create a probability distribution for this scenario.
Label Column A as “x” and Column B as “P(x).”
In Column A, list the numbers 0 to 15.
In Column B, use BINOM.DIST.RANGE to calculate the probability for each x value.
Highlight the probability cells, then right click and select Format Cells. Format the probability cells as “Number” and have Excel show 4 decimal places.
Create a probability histogram using the probabilities you calculated. Format and label it properly. Be sure to use the “Select Data” button to change the x-axis so it correctly lists the x-values.