1 Fall 2012 STAT 110 Exam 3 Practice KEY AA+B θθ+1 a1p1 + a2p2 + … + akpk2 1) A random outcome is unpredictable in the short run, but has a regular, predictable pattern in the long run. TRUE Definition of a random outcome from notes. 2) If you toss a coin 10,000 times to determine if it is fair, what type of probability are you using? Empirical You are actually observing empirical data and counting the number of times the outcome appears out of the total number of outcomes. 3) Identify the legitimate assignment of probabilities for a sample space with 4 outcomes 0.2, 0.2, 0.2, 0.4 Must all be between 0 and 1 and add to 1. A psychologist gives subjects a set of puzzles and measures how many are completed in 5 minutes. From data from many subjects, the psychologist establishes the following probability model: Puzzles solved 1 2 3 4 5 Probability ? 0.50 0.20 0.10 0.10 4) What is the probability that a subject will complete only one puzzle? 0.10 The probabilities have to add to 1 5) What is the complement of completing 3 or fewer puzzles? more than 3 puzzles The complement if an event is the opposite events (all the other outcomes). “3 or fewer” includes 1,2, or 3. Then the complement is 4 or 5. “more than 3 puzzles” means the same thing as 4 or 5 puzzles. 6) If you flip a fair coin 5 times and get 5 heads in a row, what is the probability that you will get a head on the 6th flip? 0.50 coin flips are independent so the probability of a heads on any flip is 0.5 A store hands out scratch-off tickets with cash prizes printed on them (probability model is table at right). 7) What is the expected winning for a customer? 0ሺ0.95ሻ+5ሺ0.04ሻ+100ሺ0.01ሻ=$1.20 just take each outcome times the probability and add. We do not need to subtract anything since the tickets were handed out free 8) What is the probability that a customer entering the store will not win one of the cash awards? 0.95 Not winning anything is the same as winning $0. This happens 95% of the time. 9) If several thousand customers average their winnings, we can expect their average winnings to be close to the expected value Law of Large Numbers 10) Two events are independent if The occurrence of one event does not affect the occurrence of the other event Outcome $0 $5 $100 Probability 0.95 0.04 0.013 11) TRUE or FALSE For a certain car, the events “car wins the race” and “car loses the race” would be considered disjoint TRUE cannot both happen, so disjoint 12) Suppose the odds for passing a class are 10 to 1. Find the probability of passing the class. 10/11 10 passes for every 11 classes 13) Find the probability (using a fair six-sided die) of rolling a “1” on the first roll AND rolling a “1” on the second roll. (1/6)(1/6) = 1/36 dice rolls are independent. The word “AND” tells you to multiply. The probability of rolling a “1” on a single roll is 1/6 each time, so multiply. 14) The Law of Large Numbers says that for a random sample, the sample mean will be equal to the population mean with a large enough sample size 15) What is true about the sampling distribution of the sample proportion? It has a normal distribution as long as the sample size is large A survey is being conducted using a random sample of 120 economists, asking whether paying off debt is the best usage of federal stimulus funds by the state. Of those, 33 think it is the best usage of the funds. The true percentage of all economists thinking it is the best usage is 40%. 16) Find the sample proportion, pො, of those that think paying off debt is the best use of funds. ଷଷଵଶ଴=27.5% 17) Find the mean of the distribution of the (all possible) sample proportions for those that think paying off debt is the best use of funds. 40% The mean of the normal distribution that describes all possible p-hats is the population proportion, p. 18) Find the standard deviation of the distribution of (all possible) sample proportions for those that think paying off debt is the best use of funds. ට଴.ସሺଵି଴.ସሻଵଶ଴ = ට௣ሺଵି௣ሻ௡ 19) The variability in the distribution for (all possible) pො’s will ______________ if we consider a larger sample size get smaller bigger sample size, smaller variability Wing lengths (inches) were measured from fledglings at an avian nesting site. A 90% confidence interval used to estimate the mean wing length is (2.163, 3.434). 20) Choose the correct interpretation. We are 90% confident that true mean wing length in all fledglings like these is at least 2.163 inches and at most 3.434 inches. Must use the word “confident” and not talk about the sample4 21) Suppose another confidence interval is computed with 80% confidence instead of 90% confidence. The resulting 80% confidence interval will be narrower bigger confidence level, wider interval (and vice versa) Researchers studied the effect of a houseplant fertilizer on radish sprout growth. They randomly selected some radish seeds to serve as controls, while others were planted in aluminum planters to which fertilizer sticks were added. Researchers were looking for evidence that the mean radish height is smaller when fertilized. 22) Find the HO, the null hypothesis. mean radish height is equal under fertilized or control conditions null hypothesis is that both groups are equal 23) Find the HA, the alternative hypothesis. mean radish height is smaller under fertilized condition this is what the researchers were looking for evidence of 24) Their statistical test returned a P-value of 0.002. Interpret this P-value. There is a 0.2% chance of seeing this much evidence that mean radish height is smaller when fertilized if radish height is the same under both conditions. P-value is the percent of the time that one would see this much evidence for HA if the HO were true. Taking apart the correct answer, we have There is a 0.2% chance of seeing this much evidence check that mean radish height is smaller when fertilized evidence for HA if radish height is the same under both conditions. Under the assumption H0 is true 25) If the level of significance is α = 0.05 for a test with a P-value of 0.002, then you should reject HO (find in favor of HA) this is the correct answer! Why are the others not correct? There are only