STAT1181_Patel_Kegrin_HW3_marked

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Apr 3, 2024

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STAT 1181 Assignment 3 (33 marks) Due March 7, 2024 1. An analyst wishes to determine if there is an association between the marital status and the highest academic degree. A simple random sample of 900 people provides the data shown below. (18 marks) Single Married Divorced Widowed Total Associate degree 60 115 65 45 285 Bachelor's degree 70 110 75 50 305 Master's degree 80 90 85 55 310 Total 210 315 225 150 900 a) Provide a description of the side-by-side bar graph. (2 marks) Answer-: Each Marital Status category has three bars adjacent to each other, the percentage of individuals with Associate, Bachelor’s, and Master’s degrees, respectively. The height of the bar indicates the percentage of people in each Marital Status. Percentages are measured on the vertical axis, marital statuses across horizontal axis. b) Calculate the expected frequency of the cell “single and bachelor’s degree” and interpret this value in the context of the question. (3 marks) Answer-: The expected frequency for the cell “ single and bachelor’s degree” is approximately 71.71. c) Suppose the value of chi-square statistic is 8.7067. Is there sufficient evidence for the analyst to conclude that an association exists between the marital status and the highest academic degree?

Show all steps of a chi-square test and use an appropriate decision point to support your conclusion. (3 marks) Answer-: degree of freedom = (3 - 1) * (4 - 1) = 6 Critical value of alpha = 0.05 Critical value for chi-square test is approximately 12.59. It rejects the null hypothesis as critical value is less than chi-square statistic. So, there’s not an association between marital status and the highest academic degree based on this data. d) If one person is randomly selected from the study, what is the probability that the person is married or has an associate degree? (2 marks) Answer-:P(married or associate) = P(married) + P(associate) – P(married or associate) = 0.539. e) If a randomly selected person has a master’s degree, what is the probability that the person is divorced? (2 marks) Answer= P(Divorced|Master’s) = P(Divorced and Master’s) / P(Master’s) =0.274 or 27.4% f) Suppose one person is randomly selected from the study. Are the events A = {the person is widowed} and B = {the person has a bachelor’s degree} mutually exclusive? Justify your answer using probability. (2 marks) Answer-: Event A and B are not mutually exclusive, as it is possible for someone to be in both categories as there are 50 individuals in the study who are both widowed and have a bachelor's degree. 1 g) Suppose one person is randomly selected from the study. Are the events A = {the person is married} and B = {the person has an associate degree} independent? Justify your answer using probability calculations. (2 marks) Answer-: P(A and B) = P(A) * P(B) The event are not independent as the product of the individual is not equal to the probability of the intersection. h) If two people are randomly selected from the study, what is the probability that the first person single and the second person is divorced? (2 marks) Answer=: P(1 st is single and 2 nd is divorced) = P(1 st is single) * P(2 nd is Divorced | 1 st is Single) = 0.0581 or 5.81% 2. A market researcher wishes to compare the proportions of homes with air conditioning among communities. One hundred homes were randomly selected from each community, and the

results are shown below. If the value of 2 is 3.3073, is there sufficient evidence for the market researcher to conclude that a difference exists in the proportions of homes with air conditioning among the communities? Show all steps of a chi-square test and use an appropriate decision point to support your conclusion. (3 marks) Community A Community B Community C Total Yes 48 58 38 144 No 52 42 62 156 Total 100 100 100 300 Answer-: degree of freedom is 3.3073. Chi square statistics X 2 =bserved Frequency - Expected Frequency) 2 / Expected Frequency =8.01 Now compare the critical value from chi-square distribution table and convectional significance level of 0.05. Now chi-square statistic is greater than the siginificance value so reject the HO(null hypothesis). Thus, the without specific significance level, it cannot conclude whether there is sufficient evidence to reject the null hypothesis. 3. At a fast-food restaurant, the probability that a customer orders a hamburger is 0.73, the probability that a customer orders French fries is 0.69, and the probability that a customer orders a hamburger and French fries is 0.5. (4 marks) a) Let A be the event that a customer orders a hamburger and B be the event that a customer orders French fries. Draw a Venn diagram and label it with probabilities. (2 marks) b) What is the probability that a customer orders neither a hamburger nor French fries? (1 mark) Answer-:P(neither) = 1- P(hamburger or French fries) = 0.08 or 8%. c) If a customer orders a hamburger, what is the probability that the customer also orders French fries? Give your answer to four decimal places. (1 mark) Answer-:P(French fries | Hamburger) = P(hamburger and French fries) / P(hamburger) =0.6849 or 68.94%

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4. Textbook publishers must estimate the sales of new (first edition) books. The records of one major publishing company indicate that 25% of all new books sell more than projected, 35% sell close to the number projected, and 40% sell less than projected. Of those that sell more than projected, 80% are revised for a second edition, as are 70% of those that sell close to the number projected and 50% of those that sell less than projected. (5 marks) a) Draw a tree diagram of the above situation. Make sure it shows all the related probabilities. (2 marks) Sales | ----------+---------+--------- 25% More | 35% Close | 40% Less | | | ----- --- | -------- | -------- 80% 2nd | 70% 2nd | 50% 2nd | | | ------- | -------- | -------- Not 2nd | Not 2nd | Not 2nd b) If a book goes to a second edition, what is the probability that it sold close to the number projected in the first edition? Give your answer to four decimal places. (3 marks) P(A|B) = P(B|A).P(A) / P(B) • P(A|B) is probability that a book sold sold close to the projected number given it goes to second edition • P(B|A) is the probability that a book goes to second edition given that it sold close to the projected number 0.70. • P(A) is the probability that a book sold close to the projected number 0.35. • P(B) is the probability that a book goes to second edition. If a book goes to second edition, the probability that sold close to the number projected in the first edition is 0.3798. 5. An urn contains five red balls and seven green balls. Suppose four balls are randomly selected from the urn without replacement. What is the probability of selecting at least two red balls? Give your answer to four decimal places. (3 marks) Answer-: C(n,k)= n! / k! (n−k) ! Calculate k= 2,3, and 4. The probability of getting at least red balls from the urn when four balls are randomly selected without replacement is .05758.

STAT1181_Patel_Kegrin_HW3_marked

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