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University of Toronto Scarborough
Ken Butler

STAB22 LEC05 (Covers chapter 6) ----[COURSE ANNOUNCEMENTS]- ------ Note about End-of-chapter exercises - after being asked via e-mail to post suggested end-of-chapter problems, he is now doing so. Check out his homepage at: - 2 places to get answers for these prob's - back of book (Appendix A) - MyStatLab solutions manual Mid-term - June 11 - tentative date for midterm ------------------------------------------------------------------ ---------------[CHAPTER6]----------------- [45] Recall: 2 exam scores - How can we fairly compare the exam scores to see which one is the better performance? [46] What we used to fairly compare values of varying means, SD's, units etc. are z-scores: Z-score - (ex) - x is sth that you measure (ex. score in exam) - Suppose x has: -mean = 10, - SD = 3 - Then, x-10 has -mean = 10 - 10, - but SD = 3 => if we shift the graph over by subtracting the mean, mean val. becomes 0, but the spread does not change from addition/subtraction - So, z = (x-10)/3 will have - mean = 0/3 = 0, - SD = 3/3 = 1 - You end up getting these same values for mean, SD regardless of the data when you are getting a z-score - mean = 0 - SD = 1 (ex) Let x be some val. of interest, and let our: - mean = -5 - SD = 10 Then, x - (-5) has: - mean = -5 + 5 = 0 - SD = 10 Then, (x-(-5))/10 has - mean = 0/10 = 0 - SD = 10/10 = 1 - and this expression (ie. (x-(-5))/10 = z) So whyare wegettingz-scores? - provides common basis for comparison o ex. gives means of comparing exam scores by which one is better by putting them at common scale o whenever you take away mean, bring them over and it becomes 0, and then divide by stdev, then they have stdev 1 o => now can compare fairly b/c centre and spread are same, even tho. the prev. data had diff. means & SD's - STANDARDIZATION - calculating a z-score [48] RETURNING TO THE EXAM SCORES: Which is better? Recall: Exam 1 - 67 - mean = 50 - SD = 10 Then, Exam 2 - 62 - mean = 40 - SD = 12 Then, Implication - mark of 62 is slightly better performance, relative to mean and SD - now we have a method to compare them on common grounds [49] DENSITY CURVES & NORMAL MODEL - Need a mathematical model to describe what is occurring if we want to determine how big might a typical z-score be (ex) - normal model for this curve - works for data that is roughly symmetric, no outliers - is pretty close match, b/c it has to go thro. middle of top bars, but is close, not exact - red curve = example of normal distribution model - ie. a workable approximation of real data [50] MEAN AND STANDARD DEVIATION ON NORMAL DISTRIBUTION This is an exemplary normal distribution - mean & median = 10, which is at peak - SD - 1 SD goes up to where fcn stops curving down and starts curving out - ie. an inflection point - So, suppose the inflection points are 7, 13 - Then, SD = 13 - 10 = +3 (only need to use one inflection pt to get the SD; if we used the other, we would get same answer: - SD = 7 - 10 = -3 < - - what matters is the magnitude, NOT the sign => distance from mean to inflection point is SD of normal distribution Characteristics of normal distribution - tall peak in middle - falls the same on both sides - no outliers - symmetric - b/c its symmetric and has no outliers, then the bestmeasures of centre and spread to use are mean and SD, respectively - at the peak of normal distribution is mean - but mean and median are the same in this distribution - curve never reaches 0 (ie. touching the horizontal axis) - this is a unique curve, b/c it is the only curve possible that will have mean = 10, SD = 3 [51] Z-values AND TABLE Z Table Z - lists z-val digits in the columns towards the sides and above, and the cells of the table are the proportions that correspond to certain z-val's - see pg1047-1048 of TXT Procedure - get z-score - look up z-score in table, and the cell corresponding to it gives you prop. less. (meaning: it will give you the prop. covered by the z-val's lower than the z-val we are looking at) - note that they used grams for the x-axis, but shape of dsitribution exactly thesame if we convert to z-scores and use those instead for x-axis. - there is no definite formula associated with all normal distributions - instead, have to use table to get z-scores ------------ - Note - he will provide us with this table on exam, should we need it ------------ [52] ROMA TOMATOES Information - if you plot their weights, they reveal a normal distribution shape - mean = 74g, SD = 2.5g ====== Question - what prop. (proportion, or percentage) of these tomatoes will weigh less than 70g? Solution Now, lets go look in table Z for cell that corresponds to z-value -1.60 - we get 0.0548 => to get the percentage, multiply by 100: - therefore, less than about 5.5% of the tomatoes will weigh less than 70g. ----- - Note - the prop. coming from the corresponding cell for the z-score always gives us "less" ----- [53] ROMA TOMATOES (Q2) Information - if you plot their weights, they reveal a normal distribution shape - mean = 74g, SD = 2.5g ====== Question - what prop. of these tomatoes will weigh MORE than 80g? Solution - the cell corresponding to this z-val says 0.9918 less
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