Stat Test 2
binomial experiment
A random sample of 80 middle school students is obtained, and the individuals selected are asked to state their weights? No, this probability experiment does not represent a binomial experiment because the variable is continuous, and there are not two mutually exclusive outcomes. An experimental drug is administered to 170 randomly selected individuals, with the number of individuals responding favorably recorded? Yes, because the experiment satisfies all the criteria for a binomial experiment. https://stattrek.com/online-calculator/binomial.aspx 1. is performed a fixed number of times 2. the trials are independent 3. for each trial, there are 2 mutually exclusive (disjoint) outcomes, success, or failure 4. the probability of success is the same for each trial I, IV, and VI The n trials are independent. Each trial has the same probability of a success. Each trial has two possible outcomes. The data are binary. There is the same probability of success for each trial (bid). There are a fixed number of bids. The trials are independent. You are bidding on four items available on an online shopping site. You think that you will win the first bid with a probability of 25% and the second through fourth bids with probability 30%. Let x denote the number of winning bids of the four items you bid on? No, because each trial does not have the same probability of success. You are bidding on four items available on an online shopping site. Each bid is for $70, and you think there is a 25% chance of winning a bid, with bids being independent events. Let x be the total amount of money you pay for your winning bids? No, because x does not count the number of successes. Each trial has 2 potential outcomes, but a team's previous performance affects their probability to win, and there is a lower probability of winning against the best team than against the worst team. The trials are independent and binary, but each trial may not have the same probability of success, as different size monitors could have a different rate of defect. The trials do not have the same probability and are not independent, because after each trial, the probability of selecting a female changes for the next trial. It is unlikely that each trial has the same probability of success. Voters have their own preferences, so the probability of voting for the Democratic candidate varies among the voters. It is unlikely that the trials are independent of each other, because if one family member goes to church, then the rest will go as well. They are satisfied because 1) the data are binary (Hispanic or not), 2) the probability of success is always 0.44 and 3) the trials are independent (the first selection does not affect the next; n<10% of population size). Yes, because the chance that this would occur if the selection were done randomly is very low. An investor randomly purchases 14 stocks listed on a stock exchange. Historically, the probability that a stock listed on this exchange will increase in value over the course of a year is 45%. The number of stocks that increase in value is recorded. Yes, because the experiment satisfies all the criteria for a binomial experiment. Yes, because the probability of 13 or more adults believing the overall state of moral values is poor is very low.
What is an random variable?
A random variable is a numerical measure of the outcome of a probability experiment.
Not independent events
A survey asks subjects whether they believe that global warming is happening (yes or no) and how much fuel they plan to use annually for automobile driving in the future, compared to their past use (less, same, more)? The two events are not independent. This means the probability of responding "yes" on global warming and "same" on future fuel use is less than it would be if the two choices were not related. According to a center for disease control, the probability that a randomly selected person has hearing problems is 0.148. The probability that a randomly selected person has vision problems is 0.083. Can we compute the probability of randomly selecting a person who has hearing problems or vision problems by adding these probabilities? No, because hearing and vision problems are not mutually exclusive. So, some people have both hearing and vision problems. These people would be included twice in the probability.
Independent Events of P(A and B)=P(A)xP(B)
At the local cell phone store, the probability that a customer who walks in will purchase a new cell phone is 0.07. The probability that the customer will purchase a new cell phone protective case is 0.35. Is this information sufficient to determine the probability that a customer will purchase a new cell phone and a new cell phone protective case? If so, find the probability. If not, explain why not. The information is not sufficient. The events might not be independent. In a lottery, 6 numbers are randomly sampled without replacement from the integers 1 to 48. Their order of selection is not important. Find the probability of holding a ticket that has zero winning numbers out of the 6 numbers selected for the winning ticket out of the 48 possible numbers. P(have zero of the 6 winning numbers)= 48 possible outcomes with 42 outcomes for not winning so P(42/48) For the 2nd number, there are 47 possible outcomes with 41 outcomes that are not winning numbers so P(41/47). Basically, 42/48 x 41/47 x 40/46 x 39/45 x 38/44 x 37/43=answer If E and F are disjoint events, then P(E or F)=P(E)+P(F). If E and F are not disjoint events, then P(E or F)=P(E)+P(F)-P(E and F).
What are the two requirements for a discrete probability distribution?
EP(x)=1 and 0 < equal than P(x) < equal than 1. Each probability must be between 0 and 1, inclusive, and the sum of the probabilities must equal 1.
probability
In the short run, the proportion of a given outcome can fluctuate a lot. A long run of observations is needed to accurately calculate the probability of flipping heads. Flip the coin many times to obtain a long run of observations. No. In the short run, the proportion of a given outcome can fluctuate a lot. As more people are sampled, the proportion should approach the real probability. Consider a random number generator designed for equally likely outcomes. If numbers between 0 and 49 are chosen, determine which of the following is not correct? b is incorrect because in the short run, probabilities of each integer being generated can fluctuate a lot. A pollster agency wants to estimate the proportion of citizens of the European Union who support same-sex unions. She claims that if the sample size is large enough, she does not need to worry about the method of selecting the sample. Is the pollster's statement correct? The statement is not correct. Samples should be chosen randomly to ensure that each individual in the population has about the same probability of being chosen. Before the first attempt to land on the moon, the astronaut was asked to assess the probability that he would be successful. Did he need to rely on the relative frequency definition or the subjective definition of probability? The subjective definition because the astronaut would have to use his own judgement rather than objective information such as data. Is there intelligent life on other planets in the universe? If you are asked to state the probability that there is, would you need to rely on the relative frequency or the subjective definition of probability? You would need to rely on the subjective definition because you would be relying on your own judgment. Two friends decide to go to the track and place some bets. One friend remarks that in an upcoming race, the number 3 horse is paying 60 to 1. This means that anyone who bets on the 3 horse receives $60 for each $1 bet, if in fact the 3 horse wins the race. He goes on to mention that it is a great bet, because there are only eight horses running in the race, and therefore the probability of horse 3 winning must be 1/8. Is the last statement true or false? The statement is false because there is no reason to think that all horses have an equally likely chance of winning. What is the probability of an event that is impossible? 0 Suppose that a probability is approximated to be zero based on empirical results. Does this mean that the event is impossible? No What does it mean for an event to be unusual? Why should the cutoff for identifying unusual events not always be 0.05? An event is unusual if it has a low probability of occurring. The choice of a cutoff should consider the context of the problem. In a probability model, the sum of the probabilities of all outcomes must equal 1? True Probability is a measure of the likelihood of a random phenomenon or chance behavior? True In probability, a(n) experiment is any process that can be repeated in which the results are uncertain. A(n) event is any collection of outcomes from a probability experiment. If a person rolls a six-sided die and then draws a playing card and checks its color, describe the sample space of possible outcomes using 1, 2, 3, 4, 5, 6 for the die outcomes and B, R for the card outcomes? The sample space is S={1B,2B,3B,4B,5B,6B,1R,2R,3R,4R,5R,6R}
conditional probability
The probability of being a baseball fan (BF) was 0.73 for males (M). Express this as a conditional probability. P(BF I M)=0.73 The probability of being a baseball fan (BF) was 0.49 for females (M^C). Express this as a conditional probability. P(BF I M^C)=0.49 Given is event B P(A and B)/P(B) P(carpooled to work or drove alone to work)= P(A)+P(B) P(carpooled I carpooled or dove alone to work)= P(carpooled)/P(both) Estimate the probability that the player made the second free throw, given that he made the first one. P(2nd made I given 1st made)= Both/P(1st made) Does it seem as if his success on the second shot depends strongly on whether he made the first? No because the answer to P(2nd made I given 1st made) does not equal to P(2nd made) Form a contingency table that cross classifies whether a vehicle entering the city contains radioactive material and whether the device detects radiation. Identify the cell that corresponds to the false alarms the police department fears? a b c d The cell that contains b corresponds to false alarms. Sketch a Venn diagram for which each event has similar (not the same) probability but the probability of a false alarm equals 0? b is in A. Since A contains B, P(A|B)=1. Since B is a subset of A, P(B|A)<1. The poll reported that 69% of the actively disengaged group claimed to be thriving, compared to 43% of the unemployed group. Are these percentages (probabilities) ordinary or conditional? Conditional, because the statement gives the probability of one event occurring, given that another event has occurred. Express the statement "69%" of the actively disengaged group claimed to be thriving" as a probability? P(respondent claimed to be thriving|respondent is actively disengaged)=0.
The normal curve is symmetric about its mean, μ.
The statement is true. The normal curve is a symmetric distribution with one peak, which means the mean, median, and mode are all equal. Therefore, the normal curve is symmetric about the mean, μ. The area under the normal curve to the right of μ equals one half (1/2). The histogram is not bell-shaped, so a normal distribution could not be used as a model for the variable. What happens to the graph of the normal curve as the mean increases? The graph of the normal curve slides right. What happens to the graph of the normal curve as the standard deviation decreases? The graph of the normal curve compresses and becomes steeper. The notation zα is the z-score that the area under the standard normal curve to the right of zα is α.
