Slides_PracticeClass_Week1_Annotated

Contact Information Dr. Tom Stindl Email: t.stindl@unsw.edu.au Office: Anita B Lawrence 2086 My office is on the second floor of the Anita B Lawrence Centre (formerly known as the Red Centre) in room 2086. Consultation will be available on Wednesdays at 11 am to 12 pm. Week 1 MATH2089 (Statistics) Term 3, 2023 3 / 39

Organisation: Course Schedule Week Topics Covered 1 Probability (revision); Descriptive Statistics 2 Random Variables 3 Special Random Variables 4 Sampling Distributions and the Central Limit Theorem 5 Confidence Intervals for means and proportions 6 Take a deserved break 7 Hypothesis Testing 8 Inference concerning differences in means 9 Regression analysis 10 Analysis of Variance Week 1 MATH2089 (Statistics) Term 3, 2023 6 / 39

Organisation: Practice Class Monday 2 pm to 4 pm at the Keith Burrows Theatre. Objective of the practice class: to discuss key concepts and solve some exercises to supplement the Online Lectures. The PDF “Statistics Practice Class Exercises” on Moodle provide the questions for the class. You are encouraged to attend, ask questions and participate in the discussions. Questions, questions, questions! You are strongly encouraged to complete the Online Lecture on Mobius prior to the Practice Class to get the most out of it. Week 1 MATH2089 (Statistics) Term 3, 2023 9 / 39

References Recommended textbook: Applied Statistics for Engineers and Scientists (2nd or 3rd Edition), by J. Devore and N. Farnum (Duxburry Press) Additional references: Probability and Statistics for Engineers and the Sciences (7th Edition), by J. Devore (Duxburry) Applied Statistics and Probability for Engineers (5th Edition), by D. Montgomery and G. Runger (Wiley) Week 1 MATH2089 (Statistics) Term 3, 2023 12 / 39

Warm up activity - Equally Likely Outcomes Example Suppose there are n people in the Keith Burrows Theatre. 1 Find the probability that at least two people have the same birthday. 2 Calculate the probability when n = 23 are in the theatre. Week 1 MATH2089 (Statistics) Term 3, 2023 15 / 39

Warm up activity - Equally Likely Outcomes Then by plugging in n = 23 and using MATLAB: >> 1 - prod((365-23+1):365) / 365^23 ans = 0.5073 i.e. P ( A 23 ) = 0 . 51. Week 1 MATH2089 (Statistics) Term 3, 2023 18 / 39

Warm up activity - Equally Likely Outcomes The probability that at least one person has the same birthday as you is then P ( B c n ) = 1 - P ( B n ) = 1 - ✓ 364 365 ◆ n Set P ( B c n ) = 0 . 5 to get 1 - ✓ 364 365 ◆ n = 0 . 5 = ) ✓ 364 365 ◆ n = 0 . 5 By taking the natural log n log ✓ 364 365 ◆ = log( 0 . 5 ) = ) n = log( 0 . 5 ) log ( 364 365 ) ⇡ 252 . 65 . Week 1 MATH2089 (Statistics) Term 3, 2023 21 / 39

Warm up activity - Conditional Probability Example Suppose four cards are dealt from the top of a well shuffled deck of 52 playing cards. Find the probability that all four cards are aces. The probability that the first card is an ace is 4/52. Given that the first card is an ace, the probability that the second card is an ace is 3/51. Given that the first and second card are aces, the probability that the third card is an ace is 2/50. Continuing this argument, 4 52 ⇥ 3 51 ⇥ 2 50 ⇥ 1 49 = 1 270725 . Week 1 MATH2089 (Statistics) Term 3, 2023 22 / 39

Exercise 1 If you haven’t already please read the Course Outline . Week 1 MATH2089 (Statistics) Term 3, 2023 23 / 39

Exercise 2 Did you complete Revision and Online Lecture 1 on Mobius? ! The deadline is Sunday 11 pm Week 2 Week 1 MATH2089 (Statistics) Term 3, 2023 24 / 39

Exercise 3 a) Answer the following yes/no questions, and explain your answer. (i) Will the sample mean always correspond to one of the observations of the sample? Week 1 MATH2089 (Statistics) Term 3, 2023 25 / 39

Exercise 3 a) Answer the following yes/no questions, and explain your answer. (ii) Will exactly half of the observations in a sample always fall below the mean? Week 1 MATH2089 (Statistics) Term 3, 2023 26 / 39

Exercise 3 a) Answer the following yes/no questions, and explain your answer. (iii) Will the sample mean always be the most frequently occurring data value in the sample? Week 1 MATH2089 (Statistics) Term 3, 2023 27 / 39

Exercise 3 a) Answer the following yes/no questions, and explain your answer. (iv) Can the sample standard deviation be equal to zero? Week 1 MATH2089 (Statistics) Term 3, 2023 28 / 39

Exercise 3 a) Answer the following yes/no questions, and explain your answer. (v) Can the sample median be equal to the sample mean? Week 1 MATH2089 (Statistics) Term 3, 2023 29 / 39 Yes

Exercise 3 b) (i) Suppose that you add 10 to all of the observations in a sample. How does this change the sample mean? How does it change the sample standard deviation? Week 1 MATH2089 (Statistics) Term 3, 2023 30 / 39

Exercise 3 Week 1 MATH2089 (Statistics) Term 3, 2023 31 / 39

Exercise 3 b) (ii) Suppose that you multiply all of the observations in a sample by 2. How does this change the sample mean? How does it change the sample standard deviation? Week 1 MATH2089 (Statistics) Term 3, 2023 32 / 39

Exercise 3 Week 1 MATH2089 (Statistics) Term 3, 2023 33 / 39

Exercise 3 b) (iii) A sample of temperature measurements in a furnace yielded a sample average of 446°Celsius and a sample standard deviation of 5.8°Celsius. You would like to communicate this information to an American colleague. What are the sample average and the sample standard deviation expressed in °Fahrenheit? ( Hint : temperature in °C = (temperature in °Fahrenheit - 32) ⇥ 5 / 9) Week 1 MATH2089 (Statistics) Term 3, 2023 34 / 39

Exercise 4 An experiment to investigate the survival time (in hours) of an electronic component consists of placing the parts in a test cell and running them for 100 hours under elevated temperature conditions (this is called an ‘accelerated life test’). Eight components were tested with the following resulting failure times : 75 63 100 + 36 51 45 80 90 The observation 100 + indicates that the unit still functioned at 100 hours. Is there any meaningful measure of location that can be calculated for these data? What is its numerical value? Week 1 MATH2089 (Statistics) Term 3, 2023 35 / 39

Exercise 5 Consider a sample of observations x 1 , x 2 , . . . , x n . For what value a is the quantity f ( a ) = 1 n - 1 n X i = 1 ( x i - a ) 2 minimised? Interpret in terms of the location and dispersion parameters that you know. Week 1 MATH2089 (Statistics) Term 3, 2023 36 / 39

Exercise 5 Week 1 MATH2089 (Statistics) Term 3, 2023 37 / 39

Exercise 6 Complete Revision Lecture Quiz and Lecture 1 Quiz on Mobius. Deadline: Sunday 11 pm Week 2 Week 1 MATH2089 (Statistics) Term 3, 2023 38 / 39

Any Final Questions? See you next week! Week 1 MATH2089 (Statistics) Term 3, 2023 39 / 39