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Lecture

# STAB22-LEC05-(6).docx

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University of Toronto Scarborough

Statistics

STAB22H3

Ken Butler

Fall

Description

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: http://www.utsc.utoronto.ca/~butler/b22/
- 2 places to get answers for these prob's
- back of book (Appendix A)
- MyStatLab solutions manual
http://media.pearsoncmg.com/intl/pec/mylab/XL/2012/deveaux_1ce/ssm/deveaux_1ce_ssm.html
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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