Week2-2up - today QUIZ NEXT TUESDAY!! Material covered...

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Unformatted text preview: today QUIZ NEXT TUESDAY!! Material covered through 99 – 107 For Monday: Read pages 28 (stem and leaf), 68-71, 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. £ ¢ ¢  £ ¥ § ¨¦   ¤ ¤ ¤ ¤  ¥   The Median (p. 43)  © ¡ ¡ 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   ¥ ¢  £ £   © §  §  ! "   28  ¤  §   ¢ §  ! ¡  the value of ¢ £ is below mean is above mean is away from the mean. tells us how many standard deviations ¤ " " EXAMPLE: Suppose that    8 units 2σ µ = 14 and . (new) ; (p. 56) for populations use and in place of and , Works for POPULATIONS and SAMPLES: have z-scores 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 z-score is a measure of relative standing. ¥   (p. 65) ¥   ¤ ¥  ! ¦ ¥ ¥ © ! Its population z-score is ¡ § ¡ § ¢ (p. 65)  ¤  £ §  ¤ ¥ ¡ is ¥  z-score corresponding to § ¨ is an observation from a sample, the sample ¤ § ¨ §   ¡  ¨ If ¤ © Interpretation 29  ¦  ¨ ¡ ¡ ¡  ¨  ¢ ¨ ¥  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 x-3s x-2s x-s 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 bell-shaped (mound-shaped) frequency The Empirical Rule STA 2023 c B.Presnell & D.Wackerly - Lecture 3 © © © x ¨ Table : some selected -values §    x-ks ¡ ¡ ¡ ¢ at least 1-1/k 2 ¥ £   ¢ § STA 2023 c B.Presnell & D.Wackerly - Lecture 3 ¨  ¡ ¡ ¡  ¢ ¢ !   and . and ; Not bell shaped or don’t know—use Tcheby. Bell shaped—use Emp. Rule No conflict with Tchebysheff: almost all z-scores are between approx. 95% of z-scores are between . , standard deviation ? ? the 3-bedroom homes have utility bills less than a mound-shaped 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 z-scores are between £ room homes : mean ¤ distribution (p. 65), ¡ ¥ § EXAMPLE: Survey of monthly utility bills for 3 bed For z-scores, the Emp. Rule says for a bell-shaped  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   ¤  ¢ ¥  ¥ £ ¥ ¡ £ § ¢ , or compute 36 x-s x 68% x+s (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 3-bedroom homes monthly heating ¨ ©  By Tchebysheff’s Theorem, with for 155 : ¡ ¢ ¡ ¤ for 95 : Or, compute ¡ ¤ ¡ ¥ ¥ Similarly, ¢  ¤ ¤ ¡ £ std. dev. less than . ¢ is £ That is, ¡   § ¢ ¡ § ¢ ¤ ¢ a. Notice ¢ ¤ ¤ ¢ ¥ © ¤ ¤ ¢    ¥ £ ¢ ¥ ¥ ¤ ¤ ¡ ¥ ¤  ¡ ¡ ¢ ¡ ¢  ¤ ¢ § ¡ ¡ ¢ ¤  § ¡  ¢ ¤ ¥ £ § ¥ ¥  ¥ ¡   § ¡ § ¡ ¡ ¤  ¢  ¡ ¡ § § ¥ ¥ ¡ ¥ ¥ ¥  ¥ ¥ ¥ ¤  § § ¥ ¥ ¨ ¥ ¢ £ £  ¢ ¢ ¤ ¤ § ¢ § ¥ ¤ § ¥ £ ¢ £ ¤ £ £ £ £ Solution. Know: ¤ ¥ STA 2023 c B.Presnell & D.Wackerly - Lecture 3 ¢ £ ¥ ¢ ¢ § ¢ ¤ ¡ ¤ £ ¡ ¤ ¡  ¤  ©  ¡   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) ! z-score: ¤ £ £ ¦ ¥ § ¥ £ ¥ ¥ ¥ ¦ ¤ ¤  ! £ £ ¥ . 38 -2 0 95% 2 2.5 2.5% percentile for this z-score. 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 bell-shaped ¥ £ 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 z-scores are respectively). than and § § ¥  § § 39 , The values 101 and 98 have z-scores that are even less ¥ £ ¥ corresponds to a z-score 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 bell-shaped 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 z-scores that are unusually z-score 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 Bell-shaped Not Bell-shaped 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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