TRAINING AND DEVELOPMENT C235(120 QUESTIONS AND ANSWERS) 2024 REVISED(ACTUAL EXAM) GRADED, Exams of Statistics

Download TRAINING AND DEVELOPMENT C235(120 QUESTIONS AND ANSWERS) 2024 REVISED(ACTUAL EXAM) GRADED and more Exams Statistics in PDF only on Docsity!   STATISTICS MODULE 7- PROBABILITY QUESTIONS WITH ANSWERS 2024 UPDATED LATEST GRADED 100 % PASS. Examples of Quantitative and Qualitative Probability - Answ Qualitative Description= impossible Quantitative Probability = 0% probability Qualitative Description= Unlikely Quantitative Probability = 1% to 30% probability Qualitative Description= As likely as unlikely Quantitative Probability = 40-60% Qualitative Description= likely Quantitative Probability = 70-99% Qualitative Description= certain Quantitative Probability = 100% experiment - Answ the procedure or situation in which the probability is being calculated example- rolling a dice, raining outcomes - Answ the possibilities of what can occur during the experiment, the results rolling a dice- 1,2,3,4,5,6 are outcomes event - Answ comprised of 1 or more outcomes die landing on an uneven number is an event probability of an event - Answ P(E)= probability of a certain event occuring if R= RAIN and the weatherman says there is a 40% chance of rain, then P(R)= .40 an experiment is fair if - Answ each outcome is equally likely An experiment is the procedure to test the occurrence of an event. True or False? - Answ False Correct. This is a false statement. An experiment is the procedure for which the probability of an event is calculated. set - Answ simply a collection of unique elements. For example, a set of tree species is: oak, juniper, elm, maple. subsets - Answ Set A is a subset* of set B , if every element in A is contained within B . For example: A={1,2,3} B={1,2,3,4,5} A is a subset of B , because every element in set A is contained within set B . empty set - Answ has no elements {} For example, let's say you wanted to list the days of the week that do not end in a y. There are none! Therefore, this is the empty set. In set notation, the empty set is written as a pair of brackets with nothing between them: union - Answ The union* of two sets is a collection of all of the elements listed in the sets. For example: C={2,4,6} D={1,3,5} The union of C and D is {1,2,3,4,5,6} , as those are all of the elements that appear in the sets. with a certain amount of screen time will be able to read social clues. How many surveys should the researcher collect? - Answ As many as possible. The law of large numbers holds that the more trials, the closer the empirical data comes to estimating the true probability. Correct. The answer is d. There is no way to calculate the theoretical probability in a case like this. The more data the researcher gathers the clearer an idea he or she will have about the extent of its influence (if any.) You examine the number of times a player hit a home run this season, divided by the number of at-bats the player had. What is this an example of? - Answ b) Empirical Probability Correct. The answer is b. We are using the number of times an event occurred to estimate the probability, therefore this is an example of empirical probability. sample space - Answ The sample space of flipping a coin is heads and tails. The sample space of rolling a regular six-sided die is 1 , 2 , 3 , 4 , 5 , and 6 . sample size - Answ the number of different outcomes. We need the sample space to determine the sample size, and we need the sample size to calculate probability. The sample size of flipping a coin is 2 ; the sample size of rolling a six-sided die is 6 . a list - Answ One very straightforward way to determine a sample space is to write out all the options in a list. The downside of this method is it can be difficult to determine if you have missed an outcome. To show the method, suppose someone who is planning to have two children wants to know the probability of having one boy and one girl. To calculate the probability, you first must determine the sample space of possible outcomes. This is a relatively easy sample space to find. Sample Space of Having 2 Children Boy Boy Boy Girl Girl Boy Girl Girl You can see from the table above that the sample size is 4 and there are two ways to have a boy and a girl. The probability of having 1 boy and 1 girl when having 2 children is 24=12 . When making a list, be sure to create as much of a pattern as you can to ensure you have exhausted all possibilities. For example, you might organize the list above by starting with the maximum number of boys you can get. A 20 year study creates different categories of 40 year old patients: those with a greater inherited risk of heart disease and those without; those who have healthy eating habits at the start of the study and those who do not; those who get heart disease before age 50 , after the age of 50 but before age 60 , and those who do not yet have heart disease at the end of the study. - Answ List: inherited risk-healthy eating-heart disease before age 50 inherited risk-healthy eating-heart disease between 50 — 60 inherited risk-healthy eating-no heart disease by 60 inherited risk-unhealthy eating-heart disease before 50 inherited risk-unhealthy eating-heart disease between 50 — 60 inherited risk-unhealthy eating-no heart disease by 60 no inherited risk-healthy eating-heart disease before 50 no inherited risk-healthy eating-heart disease between 50 — 60 no inherited risk-healthy eating-no heart disease by 60 no inherited risk-unhealthy eating-heart disease before 50 no inherited risk-unhealthy eating-heart disease between 50 — 60 no inherited risk-unhealthy eating-no heart disease by 60 There are 12 different categories in the study What is the size of the sample space for flipping 3 coins? - Answ From the list constructed below, there are 8 outcomes. Heads, Heads, Heads Heads, Heads, Tails Heads, Tails, Heads Heads, Tails, Tails Tails, Tails, Tails Tails, Tails, Heads Tails, Heads, Tails Tails, Heads, Heads What is the size of the sample space for rolling 1 die and flipping 1 coin? - Answ 1 , Heads 1, Tails 2, Heads 2, Tails 3, Heads 3, Tails 4, Heads 4, Tails 5, Heads 5, Tails 6, Heads 6, Tails 12 What is the size of the sample space for the order in which marbles are drawn out of a bag that contains 1 red, 1 yellow, and 1 green marble? - Answ The answer is 6. From the list that was constructed, there are 6 outcomes. Red, Yellow, Green Red, Green, Yellow Yellow, Red, Green Yellow, Green, Red Green, Red, Yellow Green, Yellow, Red What is the size of the sample space for choosing an ice cream flavor (chocolate, strawberry, vanilla) and sauce (chocolate, caramel, or none)? - Answ The answer is 9. From the list that was constructed, there are 9 outcomes. Vanilla, Chocolate Vanilla, Caramel Vanilla, None Chocolate, Chocolate Chocolate, Caramel Chocolate, None Strawberry, Chocolate Strawberry, Caramel Strawberry, None venn diagrams - Answ For example, in the real world, we might ask what is the probability that a factory worker suffers a work-related injury in the past year? This question combines being a factory worker—a machinist, a textile worker, a structural iron and steel worker, or a mechanic (among many others)—and being a person who has suffered a work-related injury —a mailman slipping as he delivers mail, an assembler suffering repetitive stress injury, a construction worker falling from scaffolding (and many other injuries). The Venn diagram* can help convey the different categories. It shows a visual representation of all the possible results. The area labeled A represents A occurring—such as someone being a factory worker in the past year—while the area labeled B represents B happening—such being a person who suffered from a workplace injury in the past year. The darker area represents both A and B occurring, that is the set of all factory workers who suffered work-related injuries last year. THE MIDDLE IS WHERE U CAN SEE HOW MANY FACTORY WORKERS SUFFERED WORK RELATED INJURIES IN THE PAST YEAR universe venn diagram - Answ In our example of factory workers in the past year ( A ) and people who have suffered from workplace injuries in the past year ( B ), the universe would be "people who were employed in the past year." The universe is defined based on the question that researchers are interested in studying. If we were calculating probabilities such as the probability of factory workers being injured at work compared to the probability of other workers being injured at work, it would make sense for the universe to be "all employed people within the past year." In the diagram below, the orange shaded area represents all the elements in the universe. "not" venn diagram - Answ "Not A " in our real-world example is "Americans employed last year as something other than factory workers." Teacher, postal worker, librarian, business woman, etc. all belong to "not A ." Note that it includes people in category B who are not in A , such as librarians who were injured by falling bookshelves. However, it is best not to explicitly consider B when trying to decide if something is not A . It's enough to ask "is this person someone who was employed last year in a type of factory work?". If not, she belongs to "not A ." In the diagram below, the orange shaded area represents all the elements in "not A ." Not B refers to outcomes that are not in the category of B but still in the universe. In our real-world example, not B would be employed people who were NOT injured at work last year, including all factory workers who were not injured at work, as well as all doctors, nurses, garbage workers, and truck drivers who were not injured at work. In the diagram below, the orange shaded area represents all the elements in "not B " a and b intersection venn diagram - Answ the middle of the venn diagram a or b union venn diagram - Answ the whole venn diagram the probabilities of each individual outcome in the sample space sum to 1. - Answ Similarly, if the experiment is flipping a coin, the events are "heads" or "tails." The probability of flipping a heads is 12 . The probability of flipping a tails is 12 . The sum of these two probabilities is 1/2 + 1/2= 1 complement - Answ the opposite of an event happening THE OCCURANCE OF THE EVENT NOT HAPPENING the sum = 1 Complementary events are those that do not have any common outcomes, and the union is the whole universe. One outcome is the event itself, and the other outcome is the event not happening. For example, flipping a coin and it landing on either heads or tails — those are the only two possible outcomes, so the two events are complementary. If there are outcomes in the universe that are not in either event, then the two events are not complementary. For example, picking either a large shirt or a medium shirt are not complementary events. There are shirts in many sizes! You could pick a large, medium, small, extra large, etc. So picking a large shirt or a medium shirt are not complementary. The events must be properly defined in order to have complementary events. For example, "rolling a die and getting an even number" would not be a complete instance of complementary events (as this is only one event). However, "rolling a die and getting an even number" and "rolling the die and getting an odd number" are complementary events; the events are clearly defined. Notice from the example above that since the sum of the probability of an event and its complement is 1 , the probability of 1 minus the probability of that event is the probability of the complement. - Answ P(A)+P(not A)=11−P(A)=P(not A)1−P(not A)=P(A) 1 − P(complement) P(at least one X) = 1 − P(no X) 2. If two events have no common outcomes, they are complementary. True of False? - Answ b. False Correct. This is a false statement. Two events with no common outcomes are complementary only if their union is the whole universe. disjoint - Answ If two events cannot both occur at the same time, they are called disjoint*. This concept applies to any number of multiple events. For example, a coin cannot be both "heads" and "tails" at the same time. The three statements below are disjoint: Examples of Disjoint Statements I was born in January. I was born in March. I was born in May. population and sample - Answ A population* and a sample* can be disjoint as well, but they shouldn't be. For example, suppose you are studying the prevalence of obesity in pre-school children. You would not survey the local high school in order to gather data about the population of pre-schoolers. That would make your population and your sample disjoint, which is nonsensical if you're trying to use your sample to learn something about your population. On the other hand, taking two disjoint samples can make subtract the intersection dependent examples - Answ The probability of picking a card that is a heart, given holding three hearts in your hand. The probability of flipping a coin and getting three heads in a row, given an initial flip of heads. The probability of getting in an accident, given a certain length of daily commute. The probability that your car is red, given that your previous car was red. The probability of getting lung cancer given a history of smoking. The probability of being a gifted athlete given that one biological parent was a gifted athlete. independent examples - Answ The probability of flipping a tails, given the previous flip was heads The probability that you are born in March, given that your spouse is born in March. The probability of drawing an ace after drawing an ace and then putting it back in the deck. conditional probability - Answ dependent events is the probability of an event occurring (such as picking the winning ticket), given that another event has already occurred (such as someone else having already picked one of the losing tickets). P (B given A)= p (A and B)/ p (A) If A and B are independent events, P(B|A)=P(B) . The probability of going fishing on a Saturday is represented as P(F) . The probability of catching a fish on a Saturday is P(C) . How would we write the probability of catching a fish on a Saturday, given that you have gone fishing? - Answ P(C|F) The General Multiplication Rule - Answ P(B∣∣∣A)=P(A and B)/ P(A) we can multiply both sides by pA to solve for p A and B P(A and B)=P(A)×P(B|A) 26% of Americans age 65 or older have diabetes. 68% of people over 65 with diabetes die from heart disease. What is the probability of having diabetes and dying from heart disease? - Answ Let A= being 65 or older with diabetes Let B= dying from heart disease after age 65 P(A)=26% P(B|A)=68% What is the P(A and B) ? P(A and B)=P(A)×P(B|A)P(A and B)=0.26×0.68=0.176 2 events are independent if - Answ P(A and B)=P(A)⋅P(B) P(A|B)=P(A) P(B|A)=P(B) P(B|A)=P(B| not A) Eight toys are being given out to eight children in a hospital ward: a green boat a blue boat a pink sparkly fairy doll a blue sparkly fairy doll a yellow toy telephone a blue whale plush animal a blue light-up star a pink light-up star - Answ t A= "getting a pink item" and B= "getting a boat." Are A and B independent events? Let us select one of the equalities above to determine if A and B are independent events. Calculate P(A|B) and P(A) . P(A|B)=0 B is given, so Ari has gotten a boat. The boats don't come in pink, so the probability that Ari gets a pink item is 0 . P(A)=1/4 There are 2 pink items and 8 items total. P(A|B)≠P(A) , therefore the events are not independent. 1/4 does not equal 0 Let A= "getting a blue item" and B= "getting a boat" Are A and B independent events? Let us use P(A and B)=P(A)⋅P(B) to test for independence. P(A and B)=18 There is one item that is both blue and a boat. The intersection of "getting a blue item" and "getting a boat" is the blue boat. P(A)=12 There are 4 blue items and 8 items total. P(B)=14 P(A)⋅P(B)=12×14=1/8 Therefore P(A and B)=P(A)⋅P(B) and the events are independent A= "getting a pink item" and let B= "getting a fairy doll" Are A and B independent events? 3) For independent events, what does P(A and B) equal? - Answ P(A)⋅P(B) A and B are independent events. P(A)=1 and P(B)=0.1. Calculate P(A and B). - Answ 0.1 A and B are independent events. P(A)=0.3 and P(B)=0.8. Calculate P(A|B). - Answ 0.3 A and B are independent events. P(A)=0.5 and P(B)=1 . Calculate P(B|A) . - Answ 1 law of total probability - Answ The fact that if you multiply across each branch and then add up the resulting products, you get 1 1. A probability tree has the same information as a tree diagram. True or False? - Answ False