Week2-4up - 30 STA 2023 c D.Wackerly - Lecture 3 STA 2023 c...

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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), 68-71, (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 z-score 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 z-scores 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 z-score is a measure of relative standing. Tchebysheff’s Theorem ¡ CIQ P ¨ ¦ (p. 65) µ = 14 ¢ RF z-score 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 bell-shaped (mound-shaped) frequency distribution, the interval (Table 2.8, p. 57): ¢ ¡ ¢   ¡ contains approximately 68% of the measurements. x+ks contains approximately 95% of the ¢  ¦¢   ¡ ¥¡ x ¢  ¤¢   ¡ £¡ x-ks  at least 1-1/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. % x-3s x-2s x-s 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 z-scores are between ¦¨ approx. 95% of z-scores are between ¦ approx. 68% of z-scores are between and © © ¦ § distribution (p. 65), 37 in 9 ¦ ¤ ¢ §  For z-scores, the Emp. Rule says for a bell-shaped 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 mound-shaped distribution, what proportion of ¡ Bell shaped—use Emp. Rule ¡ Not bell shaped or don’t know—use Tcheby. the 3-bedroom 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 3-bedroom 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 x-s 5 at least , and ¤ By Tchebysheff’s Theorem, with STA 2023 c D.Wackerly - Lecture 3 41 Example. The distribution of IQ scores is bell-shaped & D¦ ¤ 99¦ ¤ ) , . A person has an IQ of What can be said about this score? 9B¦ with . z-score:    ¨¦ ©§¥ ¦ ¤  ¨ 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 z-score. z STA 2023 c D.Wackerly - Lecture 3 43 Rare Events and Inference So, if the mean is actually ©§ £ ¦ ¤   z-score as small as -2.1. (less than 2.5% of the time!) But all three houses have z-scores 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 bell-shaped 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 z-score of ¤ $£ ¨ ¡ ¢ STA 2023 c D.Wackerly - Lecture 3 44 ¢ ¤ ¨ ¤ 4   , respectively. bedroom houses initially surveyed. B$ ¨ £ B9¦ ¤  The values 101 and 98 have z-scores that are and equipped houses are different than the standard 3- STA 2023 c D.Wackerly - Lecture 3 45 Stem-and-Leaf 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 Stem-and-Leaf Display Number of standard deviations Distribution away from the mean Bell-shaped 2 or more Not Bell-shaped 3 or more Don’t know 3 or more Stem-and-leaf 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 Stem-and-Leaf; select(double click) variable, ...
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