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Unformatted text preview: today QUIZ NEXT TUESDAY!! Material covered through 99 – 107 For Monday: Read pages 28 (stem and leaf), 6871, 2.76, 2.81, 2.83, 2.84, 2.124. Exercises: 2.47,2.64, 2.65, 2.66, 2.70, 2.71, 2.74, For tomorrow: Read pages 56–60, 64–66. Assignments : (p. 52) (p. 53) variability. Variance or standard deviation LARGE – Std. Dev. : – Variance : POPULATION – Std. Dev. : – Variance : SAMPLE MORE (p. 52) Variance and Standard Deviation The Range (p. 51) The Mode (p. 44) wrong, you have someone in mind to blame. £
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¡ Last Time: STA 2023 c B.Presnell & D.Wackerly  Lecture 3 £ Thought for the day: If you can smile when things go 27 £ STA 2023 c B.Presnell & D.Wackerly  Lecture 3 ¥
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2σ µ = 14 and . (new) ; (p. 56) for populations use and in place of and , Works for POPULATIONS and SAMPLES: have zscores between respectively. , at least standard deviations of their mean; are in the interval (p. 56) lie within of the measurements: shape of their distribution, and any For any sample of measurements, regardless of the Tchebysheff’s Theorem 6 STA 2023 c B.Presnell & D.Wackerly  Lecture 3 The size of ¤ § ¤ ¤ ¤ ! The zscore is a measure of relative standing. ¥
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STA 2023 c B.Presnell & D.Wackerly  Lecture 3 ¢ 30 ¥ ¨ 0 © x+ks 3/4 21/25
8/9 1 2 2.5 3 89% 84% 75% 0% % 31 x3s x2s xs measurements. 68%
95%
almost all x x+s x+2s contains all or almost all of the measurements. contains approximately 95% of the measurements. contains approximately 68% of the distribution, the interval (Table 2.8, p. 57): 32 x+3s For data with a bellshaped (moundshaped) frequency The Empirical Rule STA 2023 c B.Presnell & D.Wackerly  Lecture 3 ©
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¢ ! and . and ; Not bell shaped or don’t know—use Tcheby. Bell shaped—use Emp. Rule No conﬂict with Tchebysheff: almost all zscores are between approx. 95% of zscores are between . , standard deviation ? ? the 3bedroom homes have utility bills less than a moundshaped distribution, what proportion of b. If monthly utility bills for the surveyed homes have and veyed homes with monthly utility bills between a. What can you say about the percentage of all sur ¢ and ; ¢ § approx. 68% of zscores are between £ room homes : mean
¤ distribution (p. 65), ¡ ¥
§ EXAMPLE: Survey of monthly utility bills for 3 bed For zscores, the Emp. Rule says for a bellshaped
and , resp. ¡ place of in ¥ ¡ ¤ and ¢ STA 2023 c B.Presnell & D.Wackerly  Lecture 3 ¡ ¥ ¥
¢ ¥ ¢ ¥ Emp. rule works for populations too, with 33 ¢ § § § STA 2023 c B.Presnell & D.Wackerly  Lecture 3 ¡ ¡ ¡ ¡ ¡ § ¡ 34 ¢ ¡ ¡ ¥ § ¥ ¡ is ¢
¢ § ¡ ¥ § ¢ ¡ ¥ ¢ . std. dev. more than . ¡ , and ¢ costs between $95 and $155. 35 b. Note that
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(135) 16% less than $135. approx. 84% of the homes have monthly utility bills 16% Leaves approximately 32% outside. $135. , that is, between $115 and By Emp. Rule, approximately 68% fall inside the interval § STA 2023 c B.Presnell & D.Wackerly  Lecture 3 and ¢ ¤
£ at least 89% of the 3bedroom homes monthly heating ¨ © By Tchebysheff’s Theorem, with for 155 : ¡ ¢ ¡ ¤ for 95 : Or, compute ¡ ¤ ¡ ¥ ¥ Similarly, ¢ ¤ ¤ ¡ £ std. dev. less than . ¢ is £ That is, ¡ § ¢ ¡ § ¢
¤ ¢ a. Notice ¢ ¤
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£ Solution. Know: ¤ ¥ STA 2023 c B.Presnell & D.Wackerly  Lecture 3 ¢ £ ¥
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¡ ¤ £ ¡ ¤ ¡ ¤ © ¡ such that (p. 64) of the measurements are larger. upper (second) quartile = 75th percentile second quartile = median = 50th percentile lower (ﬁrst) quartile = 25th percentile percentiles: (p. 70) ! zscore:
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! £ £ ¥ . 38 2 0 95%
2 2.5 2.5% percentile for this zscore. Comment: Will see later how to ﬁnd the exact 2.5% above the 97.5th percentile. z By the Emp. Rule, can at least say this person’s score is 140 is 2.5 std. dev. above the mean. ¥ What can be said about this score? ¥ . A person has an IQ of £
¦ , ¤ with ¥ Example. The distribution of IQ scores is bellshaped
¥ £ The quartiles are just the 25th, 50th, and 75th £ of the measurements are less § STA 2023 c B.Presnell & D.Wackerly  Lecture 3 ¤
¤ percentile of a set of measurements, §
¢ , is that value of ¤ ¥ § Def. The 37 £ STA 2023 c B.Presnell & D.Wackerly  Lecture 3 ¤ ¡ ¡ ¡ ¡ ¡ ¡ § § ¤ ¢ § ¥ § § ¥ ¥ § ¤
£ ¥ § ¤ ¡ ¤ ¥ ¤
£ ¥
¥ – (their zscores are respectively). than and § § ¥ § § 39 , The values 101 and 98 have zscores that are even less ¥ £ ¥ corresponds to a zscore of
§ that those without,
§ SOLUTION: If homes with solar panels are no different might have lower utility bills? £ ¡ and . Does this suggest that solar equipped homes . Three 3 bedroom houses with solar
¥ ¥ energy panels had monthly utility bills of
£ and ¥
¥ £ an approximate bellshaped distribution with ¡ ¢ ¥ EXAMPLE : Heating costs for 3 bedroom homes have £ ¤ ¤ ¥
, then by Emp. Rule bedroom houses initially surveyed. equipped houses are different than the standard 3 Since such an event is so rare, we infer that the solar solar equipped house are different than the others. observed a very rare event, or Either: small. But all three houses have zscores that are unusually zscore as small as 2.1. (less than 2.5% of the time!) (WHY?), we should rarely see an observation with a So, if the mean is actually STA 2023 c B.Presnell & D.Wackerly  Lecture 3 ¥ Rare Events and Inference ¢ § STA 2023 c B.Presnell & D.Wackerly  Lecture 3 ¢ ¥ ¡
¡ § ¡ 40 Number of standard deviations away from the mean 2 or more 3 or more 3 or more Shape of Distribution Bellshaped Not Bellshaped Don’t know a rare event ) if: A value will be identiﬁed as “unusual” (corresponding to Rules of Thumb STA 2023 c B.Presnell & D.Wackerly  Lecture 3 41 ...
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