Sophia __ Intro to Stats Unit 3 Milestone 3.pdf/ COMPLETE SOLUTION

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Score 26/27 You passed this Milestone 21 questions were answered correctly. 1 UquNestiIonTs we3re a—nswerMed inIcLorrEectSly.TONE 3 Which of the following is a property of bino mial distributions? 2... A survey asked 1,000 people which magazine they preferred, given three choices. The table below breaks the votes down by magazine and age group. Age Below 40 Age 40 and Above The National Journal 104 200 Newsday 120 230 The Month 240 106 If a survey is selected at random, what is the probability that the person voted for "Newsday" and is also age 40 or older? Answer choices are rounded to the hundredths place. 3 Zhi and her friends moved on to the card tables at the casino. Zhi wanted to figure out the probability of drawing a King of clubs or an Ace of clubs. Choose the correct probability of drawing a King of clubs or an Ace of clubs. Answer choices are in the form of a percentage, rounded to the nearest whole number. 4 Which of the following situations describes a discrete distribution? 5 What is the probability of NOT drawing a Queen from a standard deck of 52 cards? 6 Fifty people were asked whether they were left handed. Six people answered "yes." What is the relative frequency of left-handed people in this group? Answer choices are rounded to the hundredths place. 7 Colleen has 6 eggs, one of which is hard-boiled while the rest are raw. Colleen can't remember which of the eggs are raw. Which of the following statements is true? RATIONALE The probability of choosing the hard-boiled egg is 1/6. If she cracks an egg and it is not the hard-boiled egg, then it becomes 1/5 on the next try because there are now only 5 eggs remaining and one has to be the hard-boiled egg as she did not pick it on the first try. CONCEPT Independent vs. Dependent Events 8 Two sets A and B are shown in the Venn diagram below. Which statement is TRUE? There are a total of 2 elements shown in the Venn diagram. Set A has 8 elements. Set B has 7 elements. Sets A and B have 3 common elements. RATIONALE The intersection, or middle section, would show the common elements, which is 3. The number of elements of Set A is everything in Circle A, or 8+3 = 11 elements, not 8 elements. The number of elements of Set B is everything in Circle B, or 7+3 = 10 elements, not 7 elements. To get the total number of items in the Venn diagram, we add up what is in A and B and outside, which is 8+3+7+2=20 elements, not 2 elements. CONCEPT Venn Diagrams 9 Eric is randomly drawing cards from a deck of 52. He first draws a red card, places it back in the deck, shuffles the deck, and then draws another card. What is the probability of drawing a red card, placing it back in the deck, and drawing another red card? Answer choices are in the form of a percentage, rounded to the nearest whole number. 25% 4% 22% 13% RATIONALE Since Eric puts the card back and re-shuffles, the two events (first draw and second draw) are independent of each other. To find the probability of red on the first draw and second draw, we can use the following formula: Note that the probability of drawing a red card is or for each event. CONCEPT "And" Probability for Independent Events 10 Asmita went to a blackjack table at the casino. At the table, the dealer has just shuffled a standard deck of 52 cards. Asmita has had good luck at blackjack in the past, and she actually got three blackjacks with Aces in a row the last time she played. Because of this lucky run, Asmita thinks that Ace is the luckiest card. The dealer deals the first card to her. In a split second, she can see that it is a non-face card, but she is unsure if it is an Ace. What is the probability of the card being an Ace, given that it is a non-face card? Answer choices are in a percentage format, rounded to the nearest whole number. 69% 8% 10% 77% RATIONALE The probability of it being an Ace given it is a Non-face card uses the conditional formula: Note, that in a standard deck of 52 cards, there are 12 face cards, so 40 non-face cards. Of those non-face cards, there are only 4 Aces. CONCEPT Conditional Probability 11 Using this Venn diagram, what is the probability that event A or event B occurs? 0.42 0.22 0.60 0.78 RATIONALE To find the probability that event A or event B occurs, we can use the following formula for overlapping events: The probability of event A is ALL of circle A, or 0.39 + 0.18 = 0.57. The probability of event B is ALL of circle B, or 0.21 + 0.18 = 0.39. The probability of event A and B is the intersection of the Venn diagram, or 0.18. We can also simply add up all the parts = 0.39 + 0.18 + 0.21 = 0.78. CONCEPT "Either/Or" Probability for Overlapping Events 12 The gender and age of Acme Painting Company's employees are shown below. Age Gender 23 Female 23 Male 24 Female 26 Female 27 Male 28 Male 30 Male 31 Female 33 Male 33 Female 33 Female 34 Male 36 Male 37 Male 38 Female 40 Female 42 Male 44 Female If the CEO is selecting one employee at random, what is the chance he will select a male OR someone in their 40s? 1/2 11/18 1/3 1/18 RATIONALE Since it is possible for an employee to be a male and a person in their 40s, these two events are overlapping. We can use the following formula: Of the 18 employees, there are 9 females and 9 males, so . There are a total of 3 people in their 40s, so . Of the people in their 40s, only one is male so . CONCEPT "Either/Or" Probability for Overlapping Events 13 The average number of babies born at a private hospital's maternity wing is 6 per hour. What is the probability that three babies are born during a particular 1-hour period in this maternity wing? 0.13 0.20 0.16 0.09 RATIONALE Since we are finding the probability of a given number of events happening in a fixed interval when the events occur independently and the average rate of occurrence is known, we can use the following Poisson distribution formula: The variable k is the given number of occurrences, which in this case, is 3 babies. The variable λ is the average rate of event occurrences, which in this case, is 6 babies. CONCEPT Poisson Distribution 14 Tim rolls two six-sided dice and flips a coin. All of the following are possible outcomes, EXCEPT: 1, Tails, 6 2, 8, Heads 5, 2, Tails Heads, 3, 4 RATIONALE Recall that a standard coin has two values, {Heads or Tails}, while a standard die has six values {1, 2, 3, 4, 5, or 6}. So, obtaining a 2 is possible, however the 8 is not. CONCEPT Outcomes and Events 15 Using the Venn Diagram below, what is the conditional probability of event A occurring, assuming that event B has already occurred [P(A|B)]? 0.22 0.71 0.05 0.10 RATIONALE To get the probability of A given B has occurred, we can use the following conditional formula: The probability of A and B is the intersection, or overlap, of the Venn diagram, which is 0.1. The probability of B is all of Circle B, or 0.1 + 0.35 = 0.45. CONCEPT Conditional Probability 16 Peter randomly draws a card from a deck of 24. The odds in favor of his drawing a spade from the cards are 1:3. What is the probability ratio for Peter to draw a spade? RATIONALE Recall that we can go from " " odds to a probability by rewriting it as the fraction " ". So odds of 1:3 is equivalent to the following probability: CONCEPT Odds 17 For a math assignment, Jane has to roll a set of six standard dice and record the results of each trial. She wonders how many different outcomes are possible after rolling all six dice. What is the total number of possible outcomes for each trial? 46,656 7,776 216 36 RATIONALE We can use the general counting principle and note that for each step, we simply multiply all the possibilities at each step to get the total number of outcomes. Each die has 6 possible outcomes, so the overall number of outcomes for rolling 6 die with 6 possible outcomes each is: CONCEPT Fundamental Counting Principle 18 Mark noticed that the probability that a certain player hits a home run in a single game is 0.175. Mark is interested in the variability of the number of home runs if this player plays 200 games. If Mark uses the normal approximation of the binomial distribution to model the number of home runs, what is the standard deviation for a total of 200 games? Answer choices are rounded to the hundredths place. 28.88 5.37 5.92 0.14 RATIONALE In this situation, we know: n = sample size = 200 p = success probability = 0.175 We can also say that q, or the complement of p, equals: q = 1 - p = 1 - 0.175 = 0.825 The standard deviation is equivalent to the square root of the variance. First, find the variance. The variance is equivalent to n*p*q: Now, take the square root to find the standard deviation: CONCEPT Normal Distribution Approximation of the Binomial Distribution 19 A magician asks an audience member to pick any number from 6 to 15. What is the theoretical probability that an individual chooses the number the magician has in mind? RATIONALE If we suppose that the card chosen by the magician is fixed, then there are 10 possible values, {6, 7, 8, 9, 10, 11, 12, 13, 14, or 15}, that are all equally likely. So, the probability that a specific value is chosen is: CONCEPT Theoretical Probability/A Priori Method 20 Satara was having fun playing poker. She needed the next two cards dealt to be hearts so she could make a flush (five cards of the same suit). There are 10 cards left in the deck, and three are hearts. What is the probability that the two cards dealt to Satara (without replacement) will both be hearts? Answer choices are in percentage format, rounded to the nearest whole number. 30% 60% 26% 7% RATIONALE If there are 10 cards left in the deck with 3 hearts, the probability of being dealt 2 hearts without replacement means that we have dependent events because the outcome of the first card will affect the probability of the second card. We can use the following formula: The probability that the first card is a heart would be 3 out of 10, or . The probability that the second card is a heart, given that the first card was also a heart, would be because we now have only 9 cards remaining and only two of those cards are hearts (since the first card was a heart). So we can use these probabilities to find the probability that the two cards will both be hearts: CONCEPT "And" Probability for Dependent Events 21 Joseph selects a card from a standard deck of 52 cards and then replaces it afterwards. He decides to record the total number of red cards that he selects and calculates the proportion of red cards that he has selected so far after each pick. He then constructs a graph to visualize his results. Which of the following statements is FALSE? The relative frequency of selecting a red card changes with the increase in number of selections. This is an example of the law of large numbers. The theoretical probability for selecting a red card in each pick is 0.5. For a given number of selections, we can use Joseph's graph to find the number of diamonds that he has picked so far. RATIONALE Since we are only graphing the proportion of red cards, there is no information in the sampling distribution about the actual suit of the card. CONCEPT Law of Large Numbers/Law of Averages 22 Kate was trying to decide which type of frozen pizza to restock based on popularity: pepperoni pizza or sausage pizza. After studying the data, she noticed that pepperoni flavors sold best on the weekdays and on the weekends, but not best overall. Which paradox has Kate encountered? False Negative Benford's Law Simpson's Paradox False Positive RATIONALE This is an example of Simpson's paradox, which is when the trend overall is not the same that is examined in smaller groups. Since the sale of pepperoni flavors on weekend/weekdays is larger but this trend changes when looking at overall sales, this is a reversal of the trend. CONCEPT Paradoxes 23 Select the following statement that describes non-overlapping events. To win, Jon needs a red card. He receives a Queen of Diamonds. Receiving the King of Hearts fulfills Jon's need of getting both a face card and a heart. Jon needs to roll an even number to win. When it’s his turn, he rolls a two. Jon wants a face card so he can have a winning hand, and he receives the eight of clubs. RATIONALE Events are non-overlapping if the two events cannot both occur in a single trial of a chance experiment. Since he wants a face card {Jack, Queen or King} and he got an eight {8}, there is no overlap. CONCEPT Overlapping Events 24 Three hundred students in a school were asked to select their favorite fruit from a choice of apples, oranges, and mangoes. This table lists the results. Boys Girls Apple 66 46 Orange 52 41 Mango 40 55 If you were to choose a boy from the group, what is the probability that mangoes are his favorite fruit? Answer choices are rounded to the hundredths place. 0.39 0.13 0.75 0.25 RATIONALE The probability of a person picking mangoes as his favorite fruit given he is a boy is a conditional probability. We can use the following formula: Remember, to find the total number of boys, we need to add all values in this column: 66 + 52 + 40 = 158. CONCEPT Conditional Probability and Contingency Tables 25 Dan is playing a game where he selects a card from a deck of four cards, labeled 1, 2, 3, or 4. The possible cards and probabilities are shown in the probability distribution below. What is the expected value for the card that Dan selects? 3.5 2.5 1.0 2.0 RATIONALE The expected value, also called the mean of a probability distribution, is found by adding the products of each individual outcome and its probability. We can use the following formula to calculate the expected value, E(X): CONCEPT Expected Value 26 Which of the following is an example of a false negative? Test results confirm that a woman is not pregnant. Test results indicate that a woman is pregnant when she is not. Test results indicate that a woman is not pregnant when she is. Test results confirm that a woman is pregnant. RATIONALE Since the test results indicate negatively, showing that the woman is not pregnant when in fact she is pregnant, this is a false negative. CONCEPT False Positives/False Negatives 27 A basketball player makes 80% of his free throws. We set him on the free throw line and told him to shoot free throws until he misses. Let the random variable X be the number of free throws taken by the player until he misses. Assuming that his shots are independent, what is the probability of this player missing a free throw for the first time on the fifth attempt? 0.4096 0.0016 0.08192 0.00128 RATIONALE Since we are looking for the probability until the first success, we will use the following Geometric distribution formula: The variable k is the number of trials until the first success, which in this case, is 5 attempts. The variable p is the probability of success, which in this case, a success is considered missing a free throw. If the basketball player has an 80% of making it, he has a 20%, or 0.20, chance of missing. CONCEPT Geometric Distribution © 2020 SOPHIA Learning, LLC. SOPHIA is a registered trademark of SOPHIA Learning, LLC. About Contact Us Privacy Policy Terms of Use [Show More]

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