Unformatted text preview: 30 STA 2023 c D.Wackerly  Lecture 3 STA 2023 c D.Wackerly  Lecture 3 31 Last Time:
Thought for the day: If you can smile when things go
The Range (p. 51) wrong, you have someone in mind to blame. Variance and Standard Deviation ¡ SAMPLE 2.76, 2.81, 2.83, 2.84, 2.124. ¢
¢
$
%£
"
#!¤
¦
¨
©§
¢
¤
¥£ – Std. Dev. : ¦ Exercises: 2.47,2.64, 2.65, 2.66, 2.70, 2.71, 2.74, – Variance : For tomorrow: Read pages 56–60, 64–66. £ ¨
Assignments : (p. 52) (p. 52) ¡ POPULATION &
&
$
2£
"
#1¤
&
(
¤
¦ '£ – Std. Dev. : 99 – 107 )
£ 0¨ – Variance :
For Monday: Read pages 28 (stem and leaf), 6871, (p. 53) QUIZ NEXT THURSDAY (9/12/02)!! Material covered
variability. STA 2023 c D.Wackerly  Lecture 3 Interpretation 6 is an observation from a sample, the sample
is
(p. 65) 8 units
2σ Its population zscore is lie within is (above, below) mean (p. 56) ¡ are in the interval ¡ is away from the mean. , at least standard deviations of their mean; have zscores between ; (p. 56) and . (new) Works for POPULATIONS and SAMPLES: & and in place of and , ¢ S respectively. ) for populations use 3 tells us how many standard deviations $
¤
5
&5 B
$ E¤ B ¤ C¦ ¤ )
D 4 EXAMPLE: Suppose that of the measurements: F is (above, below) mean 9A
¤ @84
3 @84
¤97
the value of shape of their distribution, and any F
5¢¨
RF ¨ The size of For any sample of measurements, regardless of the F 5&
)
6¨ ¤ 4 The zscore is a measure of relative standing. Tchebysheff’s Theorem ¡
CIQ P ¨ ¦ (p. 65) µ = 14 ¢
RF zscore corresponding to 5¢
1¨ ¤ 4
If 33 G
¦ HF 32 STA 2023 c D.Wackerly  Lecture 3 MORE 3 Variance or standard deviation LARGE through Tuesday (9/10/02) &
¤ 0¨ ¤ 4
) 34 STA 2023 c D.Wackerly  Lecture 3 STA 2023 c D.Wackerly  Lecture 3 35 The Empirical Rule
For data with a bellshaped (moundshaped) frequency
distribution, the interval (Table 2.8, p. 57): ¢ ¡
¢ ¡ contains approximately 68% of the measurements. x+ks contains approximately 95% of the ¢
¦¢ ¡
¥¡ x ¢
¤¢ ¡
£¡ xks at least
11/k 2 contains all or almost all of the measurements. Table : some selected values F F 0 F¦ ¦
£ ¨ 0% 2 3/4 75% 2.5 21/25 84% 3 8/9 89% 1 measurements. % x3s x2s xs x x+s x+2s x+3s 68%
95%
almost all 36 ¢ EXAMPLE: Survey of monthly utility bills for 3 bedroom homes : mean and ; £ and ; ¥ £¨ ¡
¡ and . . , standard deviation a. What can you say about the percentage of all surveyed homes with monthly utility bills between
© § ¡ almost all zscores are between ¦¨ approx. 95% of zscores are between ¦ approx. 68% of zscores are between and ©
© ¦ § distribution (p. 65), 37 in 9 ¦ ¤ ¢
§ For zscores, the Emp. Rule says for a bellshaped STA 2023 c D.Wackerly  Lecture 3 ©§
£ ¦ ¨¤ and , resp. ) place of and & Emp. rule works for populations too, with STA 2023 c D.Wackerly  Lecture 3 ? b. If monthly utility bills for the surveyed homes have ¥¨ a moundshaped distribution, what proportion of ¡ Bell shaped—use Emp. Rule ¡ Not bell shaped or don’t know—use Tcheby. the 3bedroom homes have utility bills less than ©
¥ ¦ § No conﬂict with Tchebysheff: ? By Emp. Rule, approximately % fall inside the ¢ than .
. ¤ 9 ¥ £ ¦ ¤ © ¦ ©
©
¥ © ¦
©
¥ © std. dev. more than 5
¢
¦ ¦¦
© ¥ © ¤ ¡ std. dev. ¥ ¢ ¡
¢ is b. Note that
or compute ¤ ©
¤ © Similarly, ¤ ¨ ¦ ¤ 4
¥
©
©
£ ¦ ¤ ¥ ¦
© is . interval , ¤ That is, and 39 ¤ ©
£ ¦ ¤ a. Notice STA 2023 c D.Wackerly  Lecture 3 38 9¦ ¤ ¢ Solution. Know: STA 2023 c D.Wackerly  Lecture 3 , that is, between $115 and $135. Leaves approximately % outside. Or, compute ¤ F
¨ ¦ ¤ £¥ ¨ ¦ ¤ £¦ ¨ ¦
£
¤
F
¥¤¦
¢
9¦
9
¤ 9 ¤ ¦ ¨ ¦ ¤ ¨ ¦ ¤ 4
©
©
©
¥ £ ©
©¢
9¦
¤ 1¨ ¤ 4
©
¤ for 95 : 16% 16%
68% for 155 : % of the 3bedroom homes monthly approx. x x+s
(135) % of the homes have monthly utility bills less than $135. 40 heating costs between $95 and $155. STA 2023 c D.Wackerly  Lecture 3 xs 5 at least , and ¤ By Tchebysheff’s Theorem, with STA 2023 c D.Wackerly  Lecture 3 41 Example. The distribution of IQ scores is bellshaped &
D¦ ¤ 99¦ ¤ )
, . A person has an IQ of What can be said about this score? 9B¦ with . zscore:
¨¦
©§¥ ¦
¤ ¨ 9 9 ¡
¡
5 ¤ 5I
$
C$ $ , is that value of such that (p. 64) of the measurements are less
of the measurements are larger. &
¤ 6¨ ¤ 4
) percentile of a set of measurements, 140 is ¤ Def. The std. dev. the mean. By the Emp. Rule, can at least say this person’s score is
the The quartiles are just the 25th, 50th, and 75th th percentile. percentiles: (p. 70) ¡ lower (ﬁrst) quartile = 25th percentile ¡ second quartile = median = 50th percentile ¡ upper (third) quartile = 75th percentile 2.5% 2.5%
95%
2 0 2 2.5 Comment: Will see later how to ﬁnd the exact
percentile for this zscore. z STA 2023 c D.Wackerly  Lecture 3 43 Rare Events and Inference
So, if the mean is actually ©§
£ ¦ ¤ zscore as small as 2.1. (less than 2.5% of the time!)
But all three houses have zscores that are unusually . Three 3 bedroom houses with solar
§ ¦ 9 ¦ § energy panels had monthly utility bills of ¦§ ¢
9 ¤ and , then by Emp. Rule (WHY?), we should rarely see an observation with a EXAMPLE : Heating costs for 3 bedroom homes have
an approximate bellshaped distribution with ©
£ ¦ § 42 STA 2023 c D.Wackerly  Lecture 3 and 5 . Does this suggest that solar equipped homes small.
Either: B9¦§ might have lower utility bills? ¡ observed a very rare event, or SOLUTION: If homes with solar panels are no different ¡ solar equipped house are different than the others. that those without, Since such an event is so rare, we infer that the solar corresponds to a zscore of ¤ $£ ¨ ¡
¢ STA 2023 c D.Wackerly  Lecture 3 44 ¢
¤ ¨ ¤ 4
, respectively. bedroom houses initially surveyed. B$ ¨
£ B9¦ ¤
The values 101 and 98 have zscores that are
and equipped houses are different than the standard 3 STA 2023 c D.Wackerly  Lecture 3 45 StemandLeaf Displays – P. 28 Given are widths (inches) of the dominant hands
(domhwdt) of 37 honors STA 2023 students. Rules of Thumb
A value will be identiﬁed as “unusual” (corresponding to From data set you will soon use in the ﬁrst project. a rare event ) if: Character StemandLeaf Display Number of standard deviations Distribution away from the mean Bellshaped 2 or more Not Bellshaped 3 or more Don’t know 3 or more Stemandleaf of domhwdt
Leaf Unit = 0.010
1
1
2
4
5
11
12
(10)
15
13
12
2
2 Graph
OK 25
26
27
28
29
30
31
32
33
34
35
36
37 £ Shape of N = 37 0
5
01
0
000000
2
5555555555
07
9
0000000000
55 StemandLeaf; select(double click) variable, ...
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 Spring '08
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 Statistics, Standard Deviation, Variance, emp, monthly utility bills

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