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Homework 11 Due Tuesday April 12 11 59pm STAT 400 Spring 2022 D Unger Each Exercise or lettered part of an Exercise is worth 5 points Homework assignments are worth 50 points Exercise 1 Several years ago ballots in Champaign Urbana contained the following question to assess public opinion on an issue Should the State of Illinois legalize and regulate the sale and use of marijuana in a similar fashion as the State of Colorado Suppose that we would like to understand Champaign Urbana s 2017 opinion on marijuana legalization To satisfy our curiosity we obtain a random sample of 120 Champaign Urbana residents and find that 87 support marijuana legalization a Construct and interpret a 99 confidence interval for p Solution A 1 100 confidence interval for p is cid 3399 cid 3080 cid 2870 cid 3495 cid 3043 cid 3548 cid 4666 cid 2869 cid 2879 cid 3043 cid 3548 cid 4667 where cid 3404 cid 3404 0 725 cid 2876 cid 2875 cid 2869 cid 2870 cid 2868 cid 3041 cid 3399 cid 3080 cid 2870 cid 3496 cid 3404 0 725 cid 3399 cid 2868 cid 2868 cid 2868 cid 2873 cid 3496 cid 4666 1 cid 3398 cid 4667 0 725 cid 4666 0 275 cid 4667 120 cid 3404 0 725 cid 3399 2 576 0 041 cid 3404 0 725 cid 3399 0 105 cid 3404 cid 4666 0 620 0 830 cid 4667 We are 99 confident that the true proportion of Champaign Urbana residents who support marijuana legalization is between 0 62 and 0 83 b Calculate and interpret the 95 confidence lower bound Solution A 1 100 lower bound confidence interval for p is cid 4678 cid 3398 cid 3080 cid 3495 cid 3043 cid 3548 cid 4666 cid 2869 cid 2879 cid 3043 cid 3548 cid 4667 1 cid 4681 cid 3041 The 95 lower bound is cid 3398 cid 3080 cid 3496 cid 4666 1 cid 3398 cid 4667 cid 3404 0 725 cid 3398 cid 2868 cid 2868 cid 2873 cid 3496 0 725 cid 4666 0 275 cid 4667 120 cid 3404 0 725 cid 3398 1 645 0 041 cid 3404 0 725 cid 3398 0 067 cid 3404 0 658 We are 95 confident that the true proportion of Champaign Urbana residents who support marijuana legalization is greater than 0 658 c Suppose that a pollster wants to estimate the true proportion of Champaign Urbana residents that support marijuana legalization to within 0 04 with 95 confidence How many Champaign Urbana residents should this pollster survey Assume the pollster has no prior knowledge about the true proportion With no prior assumption about what p could be we will use p 0 50 cid 3404 cid 4672 cid 4673 cid 4666 1 cid 3398 cid 4667 cid 3404 cid 4672 cid 4673 0 5 cid 4666 0 5 cid 4667 cid 3404 cid 3436 0 5 cid 4666 0 5 cid 4667 cid 3404 600 25 cid 4666 round up cid 4667 cid 2870 cid 3080 cid 2870 cid 2870 cid 2868 cid 2868 cid 2870 cid 2873 0 04 cid 2870 1 96 0 04 cid 3440 We would need to sample at least 601 Champaign Urbana residents in order to be 95 confident that we are within 0 04 of the true proportion who support marijuana legalization d Now assume the pollster believes that support for legalization is somewhere between 65 and 85 and they would like to estimate the true proportion of Champaign Urbana residents that support marijuana legalization to within 0 04 with 90 confidence How many Champaign Urbana residents should this pollster survey Hint Even though the assumed possible value for p falls within a range this time you ll want to choose the safest one See part b for inspiration Solution Solution Assuming the proportion is between 65 to 85 means that we don t have to use the most conservative value of 0 50 But within the given range the safest one to use will be 0 65 cid 3404 cid 4672 cid 4673 cid 4666 1 cid 3398 cid 4667 cid 3404 cid 4672 0 65 cid 4666 0 35 cid 4667 cid 3404 cid 3436 0 65 cid 4666 0 35 cid 4667 cid 3404 384 8 cid 2870 cid 3080 cid 2870 cid 2870 cid 2868 cid 2868 cid 2873 0 04 cid 4673 cid 2870 cid 3440 1 645 0 04 We would need to sample at least 385 Champaign Urbana residents in order to be 90 confident that we are within 0 04 of the true proportion who support marijuana legalization Exercise 2 In the 2020 2021 academic year many universities and colleges considered designating credit for all courses as simply pass fail Suppose the University of Illinois surveyed its students last year with the following question Should all Spring 2021 semester courses be pass no pass a Suppose that University officials wanted to estimate the true proportion of Illinois students that wanted all Spring 2021 semester courses be pass no pass to within 1 with 90 confidence How many students would need to be surveyed With no prior assumption about what p could be we will use p 0 50 cid 3404 cid 4672 cid 4673 cid 4666 1 cid 3398 cid 4667 cid 3404 cid 4672 0 5 cid 4666 0 5 cid 4667 cid 3404 cid 3436 0 5 cid 4666 0 5 cid 4667 cid 3404 6765 1 cid 4666 round up cid 4667 cid 2870 cid 3080 cid 2870 cid 2870 cid 2868 cid 2868 cid 2873 0 01 cid 4673 cid 2870 1 645 0 01 cid 3440 We would need to sample at least 6766 students in order to be 90 confident that we are within 1 of the true proportion that wanted all Spring 2021 semester courses to be pass no pass b In a huge rush suppose the University only surveyed a random sample of 120 students with 47 students answering yes to the above question Construct and interpret a 95 confidence interval for p A 1 100 confidence interval for p is cid 3399 cid 3080 cid 2870 cid 3495 cid 3043 cid 3548 cid 4666 cid 2869 cid 2879 cid 3043 cid 3548 cid 4667 where cid 3404 cid 3404 0 392 cid 2872 cid 2875 cid 2869 cid 2870 cid 2868 cid 3041 cid 3399 cid 3080 cid 2870 cid 3496 cid 3404 0 392 cid 3399 cid 2868 cid 2868 cid 2870 cid 2873 cid 3496 cid 4666 1 cid 3398 cid 4667 0 392 cid 4666 0 608 cid 4667 120 cid 3404 0 392 cid 3399 1 96 0 045 cid 3404 0 392 cid 3399 0 087 cid 3404 cid 4666 0 304 0 479 cid 4667 We are 95 confident that the true proportion of students that wanted all Spring 2021 semester courses to be pass no pass is between 0 304 and 0 479 c Using the same sample from part b construct and interpret a 95 confidence upper bound for p Solution Solution Solution A 1 100 upper bound confidence interval for p is …

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