STAT 400 Lecture AL1 Spring 2015 Dalpiaz Answers for 7 3 7 4 The sample proportion p x n where x is the number of elements in the sample found to belong to the category of interest the number of successes and n is the sample size E P p Var P p 1 p n SD P p 1 p n A large sample confidence interval for the population proportion p is p z 2 0 p 1 p n Let X have a Binomial distribution with parameters n and p Recall that X n p n p 1 p has an approximate Standard Normal N 0 1 distribution provided that n is large enough and P z 2 z 2 1 n p 1 p X n p Show that an approximate 100 1 confidence interval for p is p z 2 2 2n z 2 2 p 1 p z 2 n 4n2 2 1 z where p 2 X n n This interval is called the Wilson interval Note that for large n this interval is approximately equal to p z 2 p 1 p n X n p n p 1 p X n p 2 z2 n p 1 p 2 z 2 X2 2 n X p n2 p2 n z 2 p 2 p p p z 2 2 1 n 2 z 2 2 2 n 2 z 2 2 p z 2 2 2 p 2 p n p n z 2 2 p 2 n p2 2 p p 0 b 2 4ac 2a b a p2 b p c 0 p 1 2 a p2 b p c 0 p1 p p2 z 2 2 2 p n z 2 2 2 1 n z 2 2 4 1 n z 2 2 2 p n p 1 2 p z 2 2 2n 1 p z 2 2 2n 2 2 z 2 2 p 2n a 0 z 2 2 1 n 2 p 2 p z 2 2 n p 2 2 p z 2 2 2n 2 1 z n 2 z 4 2 4n 2 p 2 z 2 2 n p 2 p z 2 2 p 2n z 2 2 n n z 2 2 1 p z2 2 2n 4n2 2 n z2 2 2 n z z 2 2 1 z 4 2 n z2 1 Wilson interval p 2 2 p 1 p z 2 n 4n2 z 2 X z 2 2 2 Just prior to an important election in a random sample of 749 voters 397 preferred Candidate Y over Candidate Z Construct a 90 confidence interval for the overall proportion of voters who prefer Candidate Y over Candidate Z p n 749 X 397 X n p z The confidence interval 2 90 confidence level 0 53 1 645 0 10 0 53 0 47 749 397 0 53 749 p 1 p n z 1 645 2 0 05 0 53 0 03 2 0 50 0 56 Upper tail probability 0 25 z 0 005 0 0025 0 001 0 0005 0 674 0 841 1 036 1 282 1 645 1 960 2 054 2 326 2 576 2 807 3 091 3 291 99 5 99 8 99 9 50 0 20 60 0 15 70 0 10 80 0 05 90 0 025 95 0 02 96 0 01 98 Confidence level 99 2 An article on secretaries salaries in the Wall Street Journal reports Three fourth of surveyed secretaries said they make less than 25 000 a year Suppose that the Journal based its results on a random sample of 460 secretaries drawn from every category of business Give a 95 confidence interval for the proportion of secretaries earning less than 25 000 a year n 460 p 0 75 p z The confidence interval 2 0 05 95 confidence level 0 75 1 960 0 75 0 25 460 2 0 025 0 75 0 04 p 1 p n z 1 960 2 0 71 0 79 The sample size required to obtain a confidence interval for the population proportion n z 2 Always round p with specified margin of error is 2 p 1 p n Conservative Approach up p 0 50 If it is possible that p 0 50 use p 0 50 If it is not possible that p 0 50 use p the closest to 0 50 possible value of p 3 Find the minimum sample size required for the overall proportion of voters who prefer Candidate Y over Candidate Z to within 2 with 90 confidence Assume that no guess as to what that proportion might be is available Use p 0 50 90 confidence level z n 2 0 02 0 05 0 10 2 2 2 2 1 645 0 50 0 50 1691 266 p 1 p 0 02 n 1692 Round up 4 z z 0 05 1 645 A television station wants to estimate the proportion of the viewing audience in its area that watch its evening news Find the minimum sample size required to estimate that proportion to within 3 with 95 confidence if a no guess as to the value of that proportion is available 0 03 95 confidence level 0 05 0 025 2 z z 0 025 1 960 2 Use p 0 50 2 z 2 2 1 960 n p 1 p 0 50 0 50 1067 1111 0 03 Round up b n 1068 it is known that the station s evening news reaches at most 30 of the viewing audience Use p 0 30 2 z 2 1 960 n 2 p 1 p 0 30 0 70 896 3733 0 03 Round up n 897 4 If a fair 6 sided die is rolled many times then the proportion of times when 1 How many times should the 6 die be rolled for the probability to be 0 9544 that the proportion of times when 4 6 1 1 6 comes up is between and i e within of 6 30 30 30 6 comes up will eventually get very close to Use p 1 6 95 44 confidence level z n 2 1 30 0 0456 0 0228 2 z z 0 0228 2 00 2 2 2 2 00 1 5 p 1 p 1 500 6 6 30 At least 500 times
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