Unit 8 V1 Test
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Academic Magnet High**We aren't endorsed by this school
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MATHEMATICS 1210320
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Statistics
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Dec 20, 2024
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Unit 8 –Inference for Categorical Data: Chi-Square Name: _______________________ Test Multiple Choice Use the following scenario to answer questions 1 –2: Does the amount of time you watch TV in a day influence if you are labeled as “physically fit”? A recent article was published trying to determine this. They selected a single random sample of 1200 adults and asked which category of TV viewing time they fell into and were then classified as physically fit if they scored above a certain time on a fitness test. The table below shows the results. Fitness Level Physically Fit Not Physically Fit Total Daily TV Viewing Time 0 Hours 35 147 182 1 –2 Hours 101 629 730 3 –4 Hours 28 222 250 5 or more Hours 4 34 38 Total 168 1032 1200 ______ 1) Which of the following is the most appropriate inference procedure to run to answer the question if the amount of time you watch TV in a day influence if you are labeled “physically fit”?(A) Two Proportion Z Test because you are comparing if the people in the sample are labeled as “physically fit” or “not physically fit”.(B) Chi-square test for homogeneity because the proportion of physically fit individuals and not physically fit individuals are not evenly distributed across the daily TV viewing time. (C) Chi-square test for association/independence because the proportion of physically fit individuals and not physically fit individuals are not evenly distributed across the daily TV viewing time. (D) Chi-square test for homogeneity because there were four categories of daily TV viewing time that an individual could be placed into. (E) Chi-square test for association/independence because there was a single sample taken with two variables recorded. ______ 2) Which of the following DOES NOT need to be true in order to perform a chi-square test? (A) When conducting a chi-square test for independence, data should be collected using a simple random sample. (B) When conducting a chi-square test for homogeneity, data should be collected using a stratified random sample or a randomized experiment. (C) When sampling without replacement, check that n ≤ 0.10 * N for both chi-square tests. (D) All expected counts should be greater than 5 for both chi-square tests. (E) All of the above need to be true in order to perform the appropriate test.
______ 3) Is there evidence of a relationship between the highest level of education a person has obtained and their preferred recreational sport? A study took three random samples of adults of various educational levels (high-school graduate, 4-year college graduate, higher degree graduate), and asked which of the recreational sports category they are most interested in (tennis, basketball, volleyball, golf, cycling, or other). The resulting 𝜒2statistic is 12.88. Is there evidence of a difference in the distribution of educational level among recreational sports?(A) There is convincing evidence of a difference in the distribution of educational level among recreational sports at the 10% significance level but not at the 5% significance level.(B) There is convincing evidence of a difference in the distribution of educational level among recreational sports at the 5% significance level but not at the 1% significance level.(C) There is convincing evidence of a difference in the distribution of educational level among recreational sports at the 1% significance level.(D) There is not convincing evidence of a difference in the distribution of educational level among recreational sports at any significance level.(E) There is convincing evidence of a cause-and-effect relationship in educational level and recitational sports.Use the following situation for problems 4 – 5:A biologist is studying the distribution of a particular type of plant across five different habitats. The expected distribution of the plants is based on the availability of resources in each habitat. The expected ratio of plants in the five habitats (A, B, C, D, and E) is 3:2:2:2:3. Suppose the biologist collects data on the actual number of plants in each habitat finds that they contain 22, 15, 10, 12, 41 respectively. ______ 4) For a chi-square goodness-of-fit test, which of the following tables shows the correct expected number of plants in each habitat?(A) A B C D E 3 2 2 2 3 (B) A B C D E 30 20 20 20 30 (C) A B C D E 22 15 10 12 41 (D) A B C D E 6.6 3 2 2.4 12.3 (E) A B C D E 0.3 0.2 0.2 0.2 0.3
______ 5) Which of the following intervals contains the appropriate p-value for this chi-square goodness-of-fit test?(A) p > 0.10(B) 0.05 < p < 0.10(C) 0.01 < p < 0.05(D) 0.001 < p < 0.01(E) p < 0.001______ 6) A psychologist hypothesizes that scores on an aptitude test are normally distributed, with a mean of 40 and standard deviation of 5. In a random sample of 200 people, scores are observed to have the following distribution.Score Below 35 35 –40 40 –45 Above 45 Number of People 31 60 74 35 What is the 𝜒2statistic for a goodness-of-fit test?(A) 1.783(B) 0.619(C) 2.548(D) 3.220(E) 1.988______ 7) Which of the following chi-square statistics, given a specific degree of freedom, has the largest p-value?(A) 𝜒2= 12.5with df = 6 (B) 𝜒2= 12.5with df = 5 (C) 𝜒2= 8with df = 4(D) 𝜒2= 8with df = 3(E) 𝜒2= 5.5with df = 2
______ 8) Is a voter’s political party affiliation based on where they live in the state? A chi-square test for homogeneity is performed to see if there is a significant difference between the area a person lives and their political party affiliation. The follow-up analysis is given in the table below. Republican Democrat Independent North 39 25.4 7.281 17 29.9 5.565 12 12.7 0.039 South 15 21.3 1.859 30 25.1 0.973 12 10.6 0.172 East 30 28.8 0.053 31 33.9 0.241 16 14.4 0.182 West 12 20.5 3.554 35 24.2 4.839 8 10.3 0.503 Which of the following would be a correct conclusion, based on this analysis? (A) The most significant contribution to the chi-square statistic is a result of more republicans than expected being located in the North. (B) The most significant contribution to the chi-square statistic is a result of less democrats than expected being located in the West. (C) The most significance contribution to the chi-square statistic is a result of more independents than expected being located in the North. (D) More democrats than expected are located in the North and West, while less republicans than expected are located in the North and West. (E) More democrats than expected are located in the South and East, while less republicans than expected are located in the South and East. ______ 9) Which of the following is NOT true of the 𝜒2distribution? (A) The area under the 𝜒2curve is 1 (B) 𝜒2is defined only for positive values of the test statistic (C) For small degrees of freedom, the 𝜒2curve is strongly right-skewed (D) For large degrees of freedom, the 𝜒2distribution will become approximately normal (E) 𝜒2is a family of probability distributions defined by its degrees of freedom Key: Observed Expected Contribution
Use the following situation for questions 10 –12: A new antiviral medication is being tested to see if it is effective at preventing people from getting a cold. A clinical trial had 207 subjects, who did not have a cold at the start of the study, be randomly assigned to one of three treatments: a placebo pill, a pill containing 25% antiviral medication, and a pill containing 75% antiviral medication. At the end of the study, it was recorded whether the subject got a cold at any point in the duration of the study or it they did not. The results are shown in the table below.Treatment Group Placebo 25% Antiviral Med 75% Antiviral Med Got a Cold 88 48 42 Did Not Get a Cold 17 8 10 ______ 10) What would be the appropriate number of degrees of freedom in a chi-square test for homogeneity?(A) 206(B) 2(C) 3(D) 4(E) 5______ 11) Which of the following had the largest contribution to the chi-square statistic?(A) “Got a Cold” and “Placebo”(B) “Got a Cold” and “25% Med”(C) “Got a Cold” and “75% Med”(D) “Did Not Get a Cold” and “25% Med”(E) “Did Not Get a Cold” and “75% Med”______ 12) Which of the following is the appropriate conclusion when a chi-square test for homogeneity is ran?(A) Because our p-value is so large, we do not have convincing evidence that the new medication is preventing people from getting a cold.(B) Because our chi-square statistics is so small, we do not have convincing evidence that the new medication is preventing people from getting a cold.(C) Because our p-value is so small, we do have convincing evidence that the new medication is preventing people from getting a cold.(D) Because our chi-square statistic is so large, we do have convincing evidence that the new medication is preventing people from getting a cold.(E) None of the above come to the appropriate conclusion.
Use the following situation to answer questions 13 –15: You are curious about the relationship between students' modes of transportation to school and their involvement in after-school activities. You wonder if the choice of transportation (walking, biking, driving, or taking the bus) is related to whether students participate in after-school activities (“yes” or “no”). You select an SRS of 100 students and record their mode of transportation and whether they participate in after-school activities or not. (Note that darker bars are “yes”and the lighter bars are “no”). ______ 13) What is the expected count of students who walk to school and do not participate in an after-school activity?(A) 37 (B) 5.22 (C) 20.63 (D) 12.27 (E) 87 ______ 14) What is the individual component of the chi-square statistic for the cell “Driving and Yes”?(A) 1.94 (B) 16.365 (C) 22 (D) 35 (E) 1.54 “yes” to after school activities “no” to after school activities 15 22 20 25 22 15 12 25
______ 15) A chi-square test for independence is performed and the chi-square statistic is 5.772 and the p-value is 0.123. Which of the following is the correct interpretation of the p-value in the context of the problem?(A) If the distribution of after-school activity participation is the same across all modes of transportation, the probability we get a chi-square value of 5.772 or larger with 3 degrees of freedom is 0.123. (B) The probability we get a chi-square value of 5.772 if the distribution of after-school participation is the same across all grade levels, is 0.123. (C) The probability we get a chi-square value of 5.772 if after-school participation and mode of transportation are independent, is 0.123. (D) The probability that the participation of after-school activities and mode of transportation are independent is 0.123. (E) If after-school activity participation and mode of transportation are independent, the probability we get a chi-square value of 5.772 or larger with 3 degrees of freedom is 0.123. Free Response 16) The table below shows the distribution of class level (freshman, sophomore, junior, and senior) and whether they are “satisfied” or “dissatisfied” with the school lunches provided.Freshman Sophomore Junior Senior Satisfied 14 12 10 5 Dissatisfied 11 13 15 20 a) Describe how these data could have been collected so that a test for homogeneity is appropriate. b) Describe how these data could have been collected so that a test for independence is appropriate.
c) Show how to find the expected count for “Satisfied” and “Sophomore”. d) Given the table of contributions to the chi-square statistic below, which cells contributes most to the chi-square statistic? What does this mean in the context of the problem? CONTRIBUTIONS Freshman Sophomore Junior Senior Satisfied 1.37 0.30 0.01 2.69 Dissatisfied 0.95 0.21 0.00 1.87