STATS+FINAL+F2016+v1+Steph

MA T H 26367 S tati s tical M et hods F all 2016 MATH 26367 Statistics, Final Exam PROFESSOR : Stephane Lemieux DATE : Tuesday December 15, 2016 DURATION : 120 minutes STUDENT NAME : ___________________________ STUDENT NUMBER : ________________________ SCORE : ________/60 Test Rules 1. Time Allocated: 120 minutes. 2. Use of laptops is not allowed. 3. Use of books, assignments or any materials is not allowed. 4. Cell phones and mobile devices are not allowed and must be turned off and put away during the test. 5. The use of headphones or any other listening devices is not allowed. 6. All questions must be solved individually. 7. Students may not talk or communicate to each other in any way. NOTICE: Any violation of the above mentioned rules will be considered cheating. The student will be asked to leave the classroom, the test will be graded with a zero and the college will be notified. Instructions 1. Answer all questions directly on the test paper, in the space provided. 2. Show all relevant steps and rough work, not just the final answer. 3. Marks for each question are provided in brackets; plan your time accordingly. 4. This test consists of 10 pages, and is out of a total of 50 marks. 1 | P a g e

MA T H 26367 S tati s tical M et hods F all 2016 PART A: Multiple choice questions, circle the best answer. 1 mark each, 5 total. 1. Ryan’s doctor recommended he monitor his heart rate this semester when grading lab reports since they seem to be very stressful on him. The following stem-and-leaf diagram represents Ryan’s measured heart rate at different times during the semester: Stem Leaf 8 4 6 8 9 2 9 9 9 10 11 3 4 12 1 2 4 4 7 9 What is the mode of Ryan’s heart rate this semester while marking lab reports? a. 9 b. 12 c. 99 d. 113 e. 124 2. Josh receives exactly 80 emails each week. On average 20 of those emails are from students. To approximate the probability that, in a particular week, Josh would receive at least 32 emails from students, which of the following bets shows the continuity correction you should use? a. P(X > 31) b. P(X > 32.5) c. P(X > 31.5) d. P(X < 31.5) e. P(X < 32.5) 3. Phil claims that the students in his department are ABOVE average intelligence. A random sample of 25 students’ IQ scores in his department resulted in a mean IQ score of 109. The mean population IQ score is 101 with a standard deviation of 15. What type of hypothesis test would you conduct in order to make an accurate, statistical decision? a. Two-tailed test b. Right-tailed test c. Left-tailed test 4. Stephane wants to determine how much time his students spend studying each week. He randomly selects 20% of his students in EACH of his classes and asks them how long they spend studying each week. What type of sampling did Stephane use? a. Simple random sampling b. Systematic sampling c. Stratified sampling d. Cluster sampling 2 | P a g e

MA T H 26367 S tati s tical M et hods F all 2016 PART B: Short answer questions, show your work. 5 marks each, 40 total. 1. The table below lists ticket sales for several movies (in millions of $) as well as the marketing budget for each film (in millions of $). We would like to see if a marketing budget could predict movie sales. Movie Ticket Sales (Y) (in Millions) Marketing Budget (X) (in Millions) X^2 XY 1 85 5 25 425 2 105 6 36 630 3 50 2 4 100 4 130 9 81 1170 5 55 3 9 165 6 30 1 1 30 7 80 4 16 320 8 90 7 49 630 9 130 8 64 1040 10 90 5 25 450 Sum 845 50 310 4960 Average 84.5 5 31 496 (1 Marks) Calculate the slope of the regression line of ticket sales (Y) onto marketing budget (X). (1 Marks) Calculate the intercept of the regression line of ticket sales (Y) onto marketing budget (X). (2 Mark) Calculate the predicted ticket sales of a movie with a marketing budget of $5 million. (1 Mark) Notice that two different movies have a marketing budget of $5 million. Calculate each of their residuals based on your regression line. 4 | P a g e

MA T H 26367 S tati s tical M et hods F all 2016 2. The average number of goals that NHL hockey teams score is 2.67 per game. Note that theoretically a team could score any arbitrarily large number of goals in a single game. (1 mark) What type of discrete probability distribution best describes the number of goals an NHL team will score in one game? (4 marks) Calculate the chances of a particular team scoring more than 2 goals in one game. 5 | P a g e

MA T H 26367 S tati s tical M et hods F all 2016 4. Researchers want to know if tutoring actually improves students’ exam scores. The compare the midterm exam marks of as yet untutored students with their final exam marks, after having undergone weekly tutoring sessions between the midterm exam and final exam dates. The table lists their midterm and final exam scores. Midterm % Final % Difference ( d − d ) 2 Midterm Final Difference ( d − d ) 2 55 60 5 2.25 34 50 16 156.25 66 70 4 0.25 78 82 4 0.25 78 77 -1 20.25 82 84 2 2.25 85 86 1 6.25 88 89 1 6.25 92 95 3 0.25 90 90 0 12.25 (2 marks) The mean of the difference is d = 3.5 . Calculate the standard deviation of difference and the standard error of the mean difference. (2 marks) Construct a 95% confidence interval for the average difference. (1 mark) State one possible source of bias in the experiment above. 7 | P a g e

MA T H 26367 S tati s tical M et hods F all 2016 5. Based on a sample of 24 students, their average commute time to Sheridan College is 35 minutes with a standard deviation of 15 minutes. (1 mark) What is the standard error of the sampling distribution of the mean, based on this sample? (3 marks) Construct an 80% confidence interval for the average commute time. (1 mark) What proportion of students are expected to have a commute time that is longer than the highest value in your confidence interval? 8 | P a g e

MA T H 26367 S tati s tical M et hods F all 2016 7. Based on a sample of 50 students, the average starting salary of a recent graduate of the BAIS degree program is $61,000 and the sample standard deviation is $17,000. The average starting salary of all recent Sheridan College students is $40,000. (5 marks) Perform a 99% (alpha = 0.01) hypothesis to test whether there is any significant difference in the starting salary of BAISc graduates compared to all Sheridan graduates (no assumptions made on possible direction of the difference.) 10 | P a g e

MA T H 26367 S tati s tical M et hods F all 2016 8. Based on past experience, a school believes that 6% of their undergraduates will stay at the same school and enroll in a graduate program after graduating. The school has just graduated 220 undergraduate students. (2 mark) What are the mean and standard deviation of the PROPORTION of undergraduate students from this graduating class that are expected to enroll in a graduate program at the same school? (3 marks) What’s the probability that over 10% of the current graduates will enroll in a graduate program at the same school? 11 | P a g e

MA T H 26367 S tati s tical M et hods F all 2016 (1 mark) What is the predicted blood pressure of a 35 year old who weight 220 lbs? Kth Percentile: A k = nK 100 if A k is an integer, then kth percentile is Y A + Y A + 1 2 If A k is not an integer, then kth percentile is Y ⌈ A ⌉ Inter Quartile range: Q 1 = ¿ 25 th percentile Q 3 = ¿ 75 th percentile IQR = ¿ Q 3 − Q 1 Regression: β 1 = n ∑ XY −( ∑ X )( ∑ Y ) n ∑ X 2 −( ∑ X ) 2 β 0 = Y − β 1 X ^ y = β 1 X + β 0 Residual r = y − ^ y Binomial Distribution: P ( x = k ) = ( n k ) p k ( 1 − p ) n − k C ( n,k ) = ( n k ) = n! k ! ( n − k ) ! μ = np σ = √ np ( 1 − p ) Poisson Distribution: P ( x = k ) = λ k e − λ k ! μ = λ σ = √ λ 13 | P a g e mean : y = ∑ y n samplestandard deviation : s = √ ∑ ( y − y ) 2 n − 1 z − score : z = x − y σ skew : 3 ( mean − median ) standard deviation

MA T H 26367 S tati s tical M et hods F all 2016 Standard error of proportion: SE ( ^ p ) = √ ^ p ^ q n Confidence Interval for proportion: ^ p±z α / 2 ×SE ( ^ p ) ^ p±t α ×SE ( ^ p ) ,with n − 1 df Standard deviation and standard error of the mean: SD ( y ) = σ √ n SE ( y ) = s √ n Confidence Interval for mean: y ±z α / 2 ×SE ( y ) y ±t α ×SE ( y ) ,with n − 1 df Hypothesis test of mean: One sided: ( α is the probability all in one tail) Null Hypothesis : H 0 : μ 0 = μ Alternate Hypothesis: H a : μ 0 < μ ∨ μ 0 > μ Two sided ( α 2 is the probability in each tail) Null Hypothesis : H 0 : μ 0 = μ Alternate Hypothesis: H a : μ 0 ≠μ Hypothesis test of paired difference: One sided: ( α is the probability all in one tail) Null Hypothesis : H 0 : μ d = 0 Alternate Hypothesis: H a : μ d < 0 ∨ μ d > 0 14 | P a g e