a) The runner completes his next 1500 meter race in under 4 minutes: <

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1 I. Let X be the time it takes a runner to complete a 1500 meter race. It is known that for this specific runner, the random variable X has a normal distribution with mean μ = seconds and standard deviation σ = 6.5 seconds. Compute the probability that: a) The runner completes his next 1500 meter race in under 4 minutes: X P[ X < 4 min ] = P[ X < 240 sec] = P < = [ < 1.54] = P Z b) The runner takes between 245 and 252 seconds to complete his next 1500 meter race X P 245 < X < 252 = P < < = 0.77 < < 0.31 = P Z [ ] [ ] 4011 c) In the next n = 5 races the runner's average time to complete the race is under 252 seconds. X P[ X < 252 ] = P < = P[ Z < 0.69] = / 5 6.5/ 5 d) In the next n = 3 races the runner's average time to complete the race is between 245 and 254 seconds X P 245 < X < 254 = P < < = P 1.33 < Z < 1.07 = / / / 3 [ ] [ ] 7659 e) In the next n = 5 races, the runner beats the time of 245 seconds exactly 3 times. X p = P[ X < 245 ] = P < = [ < 0.77] = P Z p() 3 = ( )( 7791) = f) In the next n = 5 races, the runner beats the time of 245 seconds at least 4 times p( 4) + p(5) = ( ) ( 7791) + ( ) ( 7791) = II. Table I.1 gives scores on a university math proficiency test for n = 24 Ontario students who Table I.2 gives scores on the same university math proficiency test for m = 30 Ontario students who were required only to complete grade 12. TABLE I.1: Page 1

2 TABLE I.2: a) Construct the stem and leaf display for Math proficiency for students who had completed grade b) Construct the stem and leaf display for Math proficiency for students who had completed only grade c) Determine the mean Math proficiency for students who 70.2 d) Determine the median Math proficiency for students who had completed only grade e) Determine the median Math proficiency for students who had completed grade f) Determine the mean Math proficiency for students who had completed only grade g) Determine the inter-quartile range (IQR) of Math proficiency for students who 14.0 h) Determine the standard deviation (s) of the Math proficiency for students who Page 2

3 i) Determine the pseudo standard deviation (PSD) of the Math proficiency for students who j) Determine the standard deviation (s) of the Math proficiency for students who 9.53 k) Determine CR/4 where CR is the crude range of the Math proficiency for students who 9.25 l) Construct the box and whisker display for the Math proficiency for students who m)construct the box and whisker display for the Math proficiency for students who l) m) n) Determine 95% confidence limits for the mean Math proficiency for students who to o) Determine 90% confidence limits for the mean Math proficiency for students who to Page 3

4 p) Determine 99% confidence limits for the mean Math proficiency for students who to q) Determine 99% confidence limits for the mean Math proficiency for students who to r) Determine 99% confidence limits for the difference in the mean Math proficiency for students who had completed grade 13 with the mean Math proficiency for students who Assume the standard deviation for both groups is the same to s) Determine 95% confidence limits for the difference in the mean Math proficiency for students who had completed grade 13 with the mean Math proficiency for students who Assume the standard deviation for both groups is the same to t) Suppose the purpose of the study was to determine the mean Math proficiency for students who had completed grade 13 was higher than the mean Math proficiency for students who Assume the standard deviation of Overall Fitness of males age doesn't depend on geographical location. Determine and carry out the test for appropriate choice for the Null Hypothesis H 0 against the appropriate alternative Hypothesis H A. Use α = and α = Page 4

5 III. A medical researcher claims that one out of every ten diabetics receiving insulin develops, within five years, antibodies against the hormone, thus requiring more costly forms of medication. Assuming the researcher's claim is correct, what is the probability that: a) In a group of n = 10 diabetics receiving insulin, at most 3 develop antibodies against the hormone within five years? p ( 0) + p(1) + p(2) + p(3) = ( 0.1) (.9) + ( 0.1) (.9) + ( 0.1) (.9) + ( 0.1) (.9) = b) In a group of n = 20 diabetics, from 2 to 6 develop antibodies against insulin within five years? p 2 + p(3) + p(4) + p(5) + p(6 ( ) ) = =.3207 c) In a group of n = 100 diabetics, at least 15 develop antibodies against insulin within five years? ( 0.1) 2 (.9) 18 + ( 0.1) 3 (.9) 17 + ( 0.1) 4 (.9) 16 + ( 0.1) 5 (.9 ) 15 + ( 0.1) 6 (. 9 ) P[ X 15 ] P[ Y 14.5] = P Z = P[ Z 1.5] = d) In a group of n = 200 diabetics, from 15 to 25 develop antibodies against insulin within five years? P 15 X 25 P 14.5 Y 25.5 = P Z = P 1.30 Z 1.30 = [ ] [ ] [ ] 8064 IV. In order to test his claim(in question III.) against the alternative that his estimate is too high a random sample of n = 1580 diabetics receiving insulin where observed for a five year period. During this period x = 115 developed immunity to insulin. a) Determine and carry out the test for appropriate choice for the Null Hypothesis H 0 against the appropriate alternative Hypothesis H A. Use α = 0.05 and α = Solution: H : p 0.10 versus H : p A < ( ) 0.10( 0.90) pˆ p Test statistic z = = = = p ( 1 ) p0 n 1580 Since z <-z 0.05 = H 0 is rejected. Conclusion: Estimate is too high (α = 0.05) The same conclusion results using α = 0.01 (-z 0.01 = ) b) Determine 95% confidence limits for the percentage of diabetics developing immunity to insulin within a 5-year period to c) Determine 99% confidence limits for the percentage of diabetics developing immunity to insulin within a 5-year period to Page 5

6 d) Determine the sample size required to estimate within 0.5 of a percentage point with a 99% level of confidence, the percentage of diabetics developing immunity to insulin within a five year period. n 2 zα /2 = p p B ( 1 ) = ( )( ) = 17, 913 Page 6