ECON 221 Business Statistics Vinison Midterm Review

What does it mean if I say that there is a 25% chance there will rain tomorrow?
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When using the addition rule to find the union between A and B, why do we subtract the intersection of A and B? Shouldn't we be able to just add P(A) and P(B)? Suppose that the probability of rain is 25%, the probability of wind is 10% and P(Rain ∩ Wind) = 5%. What is the probability that rain or wind occur? What is the probability that neither occur?
Suppose that the probability that the pool will be closed is 55%. If the probability that the pool will close conditional on rain is 85%, what does that mean about the relationship between rain and pool closings? If the probability of rain is 50%, What is the probability that the pool closes and it is raining?
How is the Bayes' Theorem derived? Suppose a patient is being tested for a disease. The probability of having the disease when you are tested is 50%. Suppose that the test is 99% accurate (if the patient has the disease the test will be positive 99% of the time). There is a 2% chance of a false positive. Given that the test is positive, what is the probability that the patient has the disease?
Suppose you have a 6-sided die and each side is equally likely. A roll of 6 is a success and every other value is a failure. What is the probability of a success? Now suppose you get to re-roll once if you fail on the first roll (If you succeed, you stop). What is the probability that you will succeed on the second roll? What is the probability that you will succeed?1.) 1/6 2.) 1/6 3.) 5/6Suppose you play a lottery where there is a 0.00001 probability that you will win $10,000 and a 0.1 probability of winning $5. These two events are mutually exclusive. The ticket costs $1. What is the expected value of buying a ticket? What is the standard deviation (don't forget you might not win)?1.)Expected value of if the mean (or 3.5) of buying a ticket. 2.) .4951If you were willing to buy a ticket in problem 2, what does that say about you?That tells us that they are a risk loving consumer who is willing to take the risk even if it has a negative impact. You generalize most people.What must be true for an experiment to have a Bernoulli distribution?There are only two possible outcomes, conventionally labeled success and failure, and the probabilities of success and failure remain the same from trial to trial. Mutually exclusive, independent, ex is flipping a coin.Over the next week (7 days), the pool has a 15% chance of closing. The closings are independent from each other. What is the probability that the pool closes for 2 days? What is the probability that the pool closes 2 days or less? What is the expected number of days that the pool will close? What is the variance?1.) 0.2096 2.) .01625 3.) 1.05 closings 4.) .9447What conditions are required for a Poisson process?1.) The number of success within a special time or special interval equals any integer between zero and infinity. 2.) The number of success counted in non overlapping intervals are independent. 3.) The probability of success in any interval is the same for all intervals of equal size and is proportional to the size of the interval.In a 20 day period, a car wash had 600 customers. How many would you expect to come in a one-day period? What is the probability that the car wash gets 20 customers on a given day?30 cars a dayWhich of the following could follow Bernoulli distributions and which could follow Poisson distributions? Number of crashes on I-69 yesterday. Number of times my dog wants to go outside. Number of students who graduate within 4 years. Number of eggs a bird lays in her nest. Number of kids who throw up at Disneyworld.1. Poisson 2. Poisson 3. Bernoulli 4. Poisson 5. BernoulliIf there are 7 blue marbles and 3 red marbles in a bowl and I draw 3 of them, what is the probability that I will draw exactly 2 blue marbles?0.525. Look at CPA notes for how to solve it.Rainfall in May has a mean of 5.05 inches and a standard deviation of 1 inch. What is the probability that we get exactly 4 inches of rain in May?0Why for a continuous distribution is P(X ≥ a) = P(X > a)?It equals 1. Look at notesIf F(x) = P(X ≤ x), what is P(X ≥ x) and why? How about P(a ≤ X ≤ b)?Look at notesWhat are the characteristics of a continuous uniform distribution? Suppose the height of a tree follows the uniform distribution where the maximum height is 21 feet and minimum height is 1 foot. What is the mean? What is the standard deviation? What is the probability that a tree is between 3 and 7 feet?SD=4.47 Mean= 0 Between 3 and 7= 1/5What is the relationship between results from a standard normal distribution and the z-scores from those results?z is how many SDs x is away from zUse the z-tables (there are some on blackboard). For a normal distribution with µ = 50 and σ = 6. What is the probability of getting a result of less than 62? What about a result of more than 62? What is the 75th percentile?1.)0.0228 2.) 3.) .67When would you use an exponential distribution? Suppose that a car wash gets 50 customers per day on average. What is the probability that the time between two customers will be 1 hour or less?0.8755Suppose Y is log-normally distributed where Y = exp(X) and X is normally distributed with mean of 4 and std dev of 3. What is the probability that y is less than 90?.5636If you conducted a voter preference poll of Americans with land-lines. What sort of bias might you expect to have and why?They would have a high preference for land-lines rather than anything else because they have used it all their lives. This would be an example of section bias.2. If you wanted to learn the effect of a college degree on wages and compared people who had earned a degree and people who hadn't what sort of sampling issue might you have?You have the possibility of nonresponse bias from the people who don't have a college degree because they are busy working to get the wages they want. Ex: Union workersIn the Truman polling example, they found that the poll was biased because lower income people were not responding at the same rate as richer ones. How could they have fixed this issue?I would use stratified random sampling because it would make certain groups have a better ability to give their poll and also give better estimates from the samplingIf I wanted to study student test scores for a school district, what would a cluster sample look like? Would a cluster sample be appropriate?It would represent how good or bad that school is doing and I think it would be appropriate because it would show how the school district is doing. Cluster sampleWhat happens to se(X̄) as n gets larger? How do you interpret this intuitively?Standard Deviation/sqrn. As n goes up uncertainty goes down. If n turns into N than there is NO uncertainty.What does the Central Limit Theorem say about a sample mean? What are the required assumptions about X for the CLT to be applied?However x is distributed, n g>=30 z= xbar-Mu/(SD/sqr(n)) xbar is normal with mean, standard deviation. There are not requirements for x.If X is normally distributed with a mean of 10 and standard deviation of 3, what is the probability that a random sample of 40 observations will have a sample mean of 13 or more?z=0What should be true in order to apply the CLT to a sample proportion?normally n>=30, nP>=5 n(1-5)>=5If 5% of your products are defective, What is approximately the probability that at least 10 out of a random sample of 100 will be defective?0.011What effect does the population correction factor for the sample mean have on se(X̄)? What is se(X̄) if n=N?SD/sqr(n) sqr(N-n/n-1)