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# Statistics Notes: Week 4 and 5 Statistics 2228

ECU

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## Popular in Elementary Statistical Methods (Dr. Teresa Obuowska)

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This 3 page Class Notes was uploaded by Rebecca Weidner on Wednesday March 9, 2016. The Class Notes belongs to Statistics 2228 at East Carolina University taught by Dr. W. Teresa Obuchowska in Spring 2016. Since its upload, it has received 24 views. For similar materials see Elementary Statistical Methods (Dr. Teresa Obuowska) in Mathematics (M) at East Carolina University.

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Date Created: 03/09/16

Elementary Statistical Methods Notes: Week 4 and 5 (Test 2 Material ONLY!!!) Feb 5 –Feb 12 Section 3.4: Measures of Position and Outliers Sample Z-score = x - xx̄ / s Population Z-score = x - mu / sigma In other words: The z-score is the specific measurement from the data set minus the mean value and then divided by the standard deviation. If a Z-score is between -2 and 2 , i.e. |z-score| < or = 2, then the value of the measurement is highly likely. If this same thing is true except the number is 3, then the measurement is highly unlikely and such an unusually large or small measurement is an outlier. *This just means that any measurement that is outside of three deviations from the mean is considered an outlier and very unlikely. Percentile – If a set of measurements on a given variable x has been ordered in ascending order, then the k-th percentile (denoted P ) is the k value of x that exceeds k% of the measurements and is less than the remaining (100-k)%. Lower Quartile (1 Quartile) – denoted Q , is th1 value of x that exceeds 25% of measurements and is less than the remaining 75%. Upper Quartile (3 Quartile) – denoted Q , is th3 value of x that exceeds 75% of measurements and is less than the remaining 25%. Q 1 P 25 Q3= P 75 Q 2 P 50= The Median! So it is less than 50% and more than the remaining 50% of measurements. How to Calculate the k-th percentile: 1. location of the k-th percentile – L = (k/100)(n+1) a. Next calculate I.F, where I is an integer part of L and Fkis the fractional part. Example: If L =k17.65, then I = 17 and F = 0.65 2. The k-th percentile : P = k + F(IX ( I + F)I Example: Find Q , Q1and3P (the 660percentile) for the following set of data: 4, 8, 9, 11, 11, 13, 16, 18, 20, 25 Sol. Position of Q :1L =25.25(11) = 2.75 so, I = 2 and F = 0.75 Q = X + 0.75( X – X ) = 8 + 0.75(9 – 8) = 8.75 1 2 3 2 Position of Q 3 L (75) = 8.25 => I = 8 and F = 0.25 Q = P = X +0.25(X – X ) = 18 + 0.25(2) = 18.50 3 75 8 9 8 P60 L 60 = 0.6(11) = 6.6 => P 60 = X 6 0.6(X – X7) =>613+0.6 = 14.8 The Interquartile Range (IQR) is defined as : IQR = Q – Q 3 1 A measurement is called an outlier if it is either: x< Q –11.5(IQR) = Lower Fence x> Q + 1.5(IQR) = Upper Fence 3 Example 2: Find the lower and upper quartiles the median, the IQR, and fences for this data: 0, 1, 5, 6, 7, 8, 9, 10, 12, 12, 13, 14, 16, 19, 21 n=15 Sol: Position of Q : L = 0.25(16) = 4 => Q = P = X = 6 1 25 1 25 4 Position of Q :2L =50.50(16) = 8 => Q = P = X =210 =50he 8 Median Position of Q :3L =75.75(16) = 12 => Q = P = X =314 75 12 IQR = Q – Q = 14-6 = 8 3 1 Lower Fence : Q – 115(IQR) = 6-12 = -6 Upper Fence : Q + 1.5(IQR) = 14+12 = 26 so, NO OUTLIERS 3 8. The average young man is 69.6 inches tall, with standard deviation sigma = 3.0 in, while the average young woman is 64.1 in. tall, with sigma = 3.8 inches. Who is relatively taller, a 67 in. man, or a 62 in. woman? Sol. Man: mu = 69.6, sigma = 3 => 67 in. man : z-score 67 – 69.6/3 = -0.87 Woman : mu = 64.1, sigma = 3.8 => 62-64.1/3.8 = -0.55 The woman is relatively taller. 18. The amount of time a student spends on homework online assignment has the following characteristics: Q 1 42 ; Q = 52.5 ; Q = 72.53min (a)Provide an interpretation of these results. (b)Determine and interpret the IQR Sol. IQR = Q – 3 = 7215 – 42 = 30.5 (c)Suppose a student spent 2 hours( 120 mins) doing HW, would this be an outlier? Is 120 > Q + 3.5(IQR) = 72.5 + 1.5(30.5) => 118.25 so, YES Review: Lk= k/100(n+1) = I.F If k = 20.65 => I = 20 and F = 0.65 Pk= X =I+ F(X I+1+ X)I P k X 20+ 0.65(X 21– X 20 Section 3.5: The five-number summary and the box-plot The five-number-summary consists of : X min, Q 1 Q ,2Q , 3 max To construct a box-plot: 1. Calculate Q , 1 , Q2and 3he IQR for the data set. 2. Draw a horizontal axis representing the scale of measurements and form a rectangle just above the horizontal line with endpoints at Q an1 Q . 3 Modified Box plot (includes fences): Lower Fence: Q – 312(IQR) Upper Fence: Q + 332(IQR) Adjacent values are the largest and the smallest measurements within fences. Example: Construct a box-plot for this data set and identify any outliers: 12, 18, 22, 23, 24, 25, 25, 26, 26, 27, 28 Sol. Position of Q :1L 25 : 0.25(12) = 3=> Q = X 1 22 3 Q 2 L 50: 0.5(12) = 6 => Q = X 2 25 6 Q : 0.75(12) = 9 => Q = X = 26 3 3 9 IQR = Q – 3 = 21 – 22 = 4 Lower fence: Q – 312(4) = 22 – 6 = 16 Upper fence: Q + 332(4) = 26 + 6 = 32

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