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# Math 2040 Chapter 4 notes Math 2040

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This 9 page Bundle was uploaded by Ang Judd on Saturday January 30, 2016. The Bundle belongs to Math 2040 at Southern Utah University taught by Said Bahi in Winter 2016. Since its upload, it has received 77 views. For similar materials see Business Statistics in Math at Southern Utah University.

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Date Created: 01/30/16

Chapter 4 4.1 Measures of Central Tendency Objective: compute values to summarize the set Numerical Descriptive Statistics: Numerical summaries of the data Arithmetic Mean (Simple Mean) 1 (1 +2 +….n ) or ẍ=∑x n n ex) Sample {5, 12, 6, 14} Find ẍ 5+12+6+14 4 ẍ= 9.25 Weighted Mean ẍw= W 1 1W X2 2.+W X n n or ẍw= ∑(WXi i W 1W +2.+W n ∑(Wi) W (weight) X ex) Table 4.4 Unemployment Rates W 1 X1 ẍw= 422.4(8.4)+2972.7(7.4)+4490.6(9.2)+9658.4(8.3)+6362(8.6) W X 2 2 422.4+2972.7+4490.6+6362 W 3 X3 w ≈ 8.24% W X * Put the variable to weigh in the calculator first*!!!!X's 4 4 . . FIRST!!!! . . ex) Walter's Grades . . ẍ = 83(40)+98(20)+90(10)+87(30) w W n Xn Homework 20% 40+20+10+30 Tests 40% ẍw= 87.9 Quizzes 10% > X's (Walter's scores listed first)< Final 30% Trimmed Mean Trimmed Mean: Ignores an equal percentage of the highest and lowest data values in calculating the mean ex) Find the average hourly wage {$5, $6, $800} ẍ= 5+6+800 = 81 = 270.3333 *$800 is an outlier* 3 3 *$270 is oviously not the average. THIS IS WHY WE NEED THE TRIMMED MEAN* ex)Find the 10% trimmed mean n=10 10% of n =10x 0.1 *Data must be ordered* Remove one observation from each side 15, 21, 25, 31, 35, 42, 48, 51, 54, 60= 307 =38.375 8 8 Median Median: The middle of an ordered array (same number of data values on either side). 50% 50% Low Median High ex) {2, 4, 0, 0, 1, 8, 5, 1, 5, 9} *order data values! 0, 0, 1, 1, 2, 4, 5, 5, 8, 9* Location of Median n+1 2 10+1 =5.5 The median is halfway between the 5th and 6th values 2 5th values is 2 6th value is 4 median is 3 Variation/ Deviation Variation of x is x-ẍ ex) {5,6,8} ẍ≈ 6.3 Overall Variation x=5 --> 5-6.3= -1.333 ∑(x-ẍ)= (5-6.333)+(6-6.333)+(8-6.333) =0.001 x=6 --> 6-6.3= -0.333 Total Deviation ∑(x-ẍ) x=8 --> 8-6.3≈ 1.677 Mode Mode: The most frequent value (or observation) ex) {1, 2, 5, 3, 1, 3, 1} X f *Useful when you cannot arrange data (nominal data)* 1 3 2 1 3 2 5 1 The Right Measure of Center Qualitative Quantitative Nominal Ordinal Interval Ratio Mean X X Median X X X Mode X X X X Trimmed Mean X X Moving Average *Used for time series* ex) Observing in time change in revenue for a business month return 2 period MA 3 period MA 1 5% 2 1% (5+1)/2 6% 3 3% (1+3)/2 2% (5+1+3)/3 3% 4 12% (3+12)/2 75% (1+3+12)/3 5% 5 11% (12+11)/2 11.50% (3+12+11)/3 8.60% 6 15% (11+15)/2 13% (12+11+15)/3 12.60% 4.2 Measures of Disperion Range, Variation, and Standard Deviation all measure the variance of data around the mean. ex) Hospital; mean weight of newborns is 3,000gm ẍ= 3,000 x = 2,500 1 x2= 4,indicates a dispersion *We want a measure fo the overall dispersion* Range Range: The highest observation - the lowest observation *Not a very good measure because it is affected by outliers* *Gives a good first look, but it's not very efficient* Standard Deviation Standard Deviation: A measure of how much we might expect a typical member of the data set to differ from the mean. Sample Mean ẍ= ∑x (statistic) n Population Mean ᵞ= ∑x (parameter) n *Sample; Latin (X,Y,Z) Population; Greek (∑,ᵞ )* Sample Standard Deviation Population Standard Deviation s=√ ∑ (x-ẍ) Ợ=√ ∑ (x-μ) n-1 n ex) Find the SAMPLE deviation {4, 10, 9, 11, 9, 7} 2 2 2 2 2 2 2 s=√ ∑ (x-ẍ) s=√(4-8.333) +(10-8.333) +(9-8.333) +(11-8.333) +(9-8.333) +(7-8.333) n-1 6-1 ẍ= 8.333 s=√18.775+2.7789+0.44489+7.1129+0.44489+1.7769 On Calculator 5 L1 L2 L3 s=√31.333 2 X X-ẍ (X-ẍ) 5 s=√6.2267 s= 2.503 Variance Variance: Standard Deviation Squared; Distance of all the values Samp2e Variance 2 Po2ulation Var2ance s = ∑ (x-ẍ) Ợ = ∑ (x-ᵞ) n-1 n ex) Find the sample variance {4, 10, 9, 11, 9, 7} 2 s = 31.333 = 6.2667 6-1 2 *Standard Deviation =√Variance Varience= Standard Deviation * Mean Deviation *Used a lot in finance, but not really anywhere else* ∑ l x-ẍ l ex) Find the MD of the sample {3, 9, 7, 8, 5} ẍ=6.4 n l3-6.4l+l9-6.4l+l7-6.4l+l8-6.4l+l5-6.4l 5 3.4+2.6+0.6+1.6+1.4 10 5 5 MD= 1.92 On the calculator L1 L2 L3 x x-ẍ lx-ẍl (L1-6.4) abs(L2) math-->num-->abs( ex) Find the population Mean Deviation and Variance {3, 9, 7, 8, 5} Ợ=√ ∑ (x-ᵞ)2 = 2.154 n Ợ = ∑ (x-ᵞ) 4.64 n Application of the Mean & Standard Deviation Empirical Rule: Use if the data set is symetrically bell shaped 68% of the values are within one standard deviation range: ẍ+/-sor μ+/-Ợ 95% of the values are withing two standard deviations range: ẍ+/-2s or μ+/-2Ợ 99.7% of the data values are within three standard deviationsrange: ẍ+/-3s or μ+/-3Ợ EX) 4.2:21b In what range can the manager expect the daily sales to be in 95% of the time? 95%= ẍ+/-2s average sales of diner ẍ= $4,500 4,500+2(750)=6,000 S= $750 4,500-2(750)=3,000 x= daily sales The manager can expect his daily sales to be between $3,000 and $6,000 95% of the time. * We assumed the data was bell shaped* Chebysher's Rule: Use if the data is not bell shaped For k>0, the proportion of values that lie within k standard deviation from the mean is at least 1-1/k2 2 1-1/k *k is the number of standard deviations* ex) if k=2 2 1-1/2 = 1-1/4 = 3/4 at least 75% of the values lie in the ex) if k=3 2 range ẍ+/- 2s 1-1/3 = 1-1/9 = 8/9 at least 88.9% of the values lie in the ex) if k=1.5 range ẍ+/- 3s 1-1/(3/2) 2 = 1-1/(9/4) = 5/4 at least 55.6% of the values lie in the range ẍ+/- 1.5s EX) 4.2:19 The average number of calls to a call center is 972 calls, with a standard deviation of 127. State the range of at least 75% of the data (calls). * We don't know if the data are bell shaped* We know k=2 --> 1-1/k =75%. The range is ẍ+/- 2s ẍ+ 2s= 972-2(127)=718 The call center will receive between 718 and 1,226 calls 75% of ẍ- 2s=972+2(127)=1,226 the time. 4.3 Measures of Relative Positions Percentiles and Quartiles *Values must be ordered* Finding the Percentiles L =n p L= location 100 n= sample size p= percentile we want If the formula L=n (p/100) results in a decimal for L, then the location of the percentile is the next whole number. ex) if n=135 & p=10 L= 135(10/100) 135 x 0.1=13.5--> 14 the 10th percentile is the value at the 14th location of the data set If the formula L=n(p/100) results in a whole number, then the location of the percentile is the average of that number and the next. n=135 p=20 L= 135(20/100)=27 The 20th percentile is the average of the 27th and 28th values Quartile Box Plot Q1: 25= n(25/100) Q : L = n(50/100) 2 50 Q3: 75= n(75/100) IQR: Inner Quartile Range Inner Quartile Range: A measure of dispersion which describes the range of the middle 50% of the data. * When the mean is not a good measure of deviation we use IQR* IQR= Q 3Q 1 ex) Q3= 25.3 and Q 1 19.8 25.3-19.8= 5.5 IQR= 5.5 Box and Whisker Plot *Easy, but quite powerful when comparing different populations* Steps 1) Order data 2) Put in minimum value and maximum value; Make sure you have even spacing 3) Plug in quartiles 4) Create a box over your number line 5) Add the whiskers to the minimum and maximum EX) 4.3:12 Construct a box plot Outliers Outliers: Kind of extreme values Upper Outlier: value > Q3+ 1.5(IQR) Lower Outlier: value< Q 11.5(IQR) EX) 4.3:12 data Upper Outlier 84+1.5(44)= 150 No upper outlier in the 4.3:12 data Lower Outlier 40-1.5(44)= -26 No lower outlier in the 4.3:12 data Z-Score Z-Score: Measures how many standard deviations a value is from the mean (Z-value) population x-ᵞ sample x-ẍ Ợ s EX) 4.3:12g Find the z-score of 81 x=81 Z= 81-61.866667 23.15127662 Z≈ 0.83 *Don't round in the middle of a calculation, round at the end* *Negative z-scores are below the average* 4.5 The Coeffient of Variation ex) 3.14 x= amount of rainfall y= price of land Which data set/ variable has the largest variation? x(rainfall y(price) ẍ= 26.08" ÿ= $117,000 * You can't compare inches (") to dollars ($)* sx= 7.55" sy= $42,931 Define the Coefficient of Variation Population CV= Ợ x100 Sample CV= S x100 ᵞ ẍ ex)3.14 CVx= 7.55 x100 CV y 42,931 x100 26.08 117,000 CV x 28.9% (rainfall) CVy= 36.7% (prices) *This data set has more variability* 4.6 Analyzing Grouped Data Finding the Mean of Grouped Data EX) 4.17 (table 4.18) Approximate (find) the mean and standard deviation of cash. X= cash on hand (sample of companies) x f midpoint (x) 1) Find 0-10mil 10 5mil 10-20mil 7 15mil x= midpoint of class 20-30mil 10(7)+7(15)+255)il(35)+1(45)+4(55)+2(65)+2(75)+2(85)+3(95) 30-40mil 7 35mi l0+7+7+7+1+4+2+2+2+3 40-50mil 1 45mil Sample Population *Same 50-60mil 4 55mil ẍ= ∑(fx) μ= ∑(fx) 60-70mil 2 65mil ∑(f) ∑(f) 70-80mil 2 75mil 80-90mil 2 85mil 90-100mil 3 95mil n=45 Finding the Standard Deviation of Grouped Data Sample s=√ ∑(fx) Population Ợ=√ ∑(fx) ∑(f) ∑(n) Finding the Variation of Grouped Data Varience = s or =Ợ2 In the calculator X 1n L Weight/Frequency 2n L FrequencyLis2 is L 4.7 Proportions n= Sample size Studying a population characteristic (trait) ex) want the proportion of left-handed in a population (how many left:right handed golf-clubs) P-parameter Define the sample population μ, Ợ Ṕ= x n x= count of objects showing the studied characteristic ex) n=447 (sample size) x=89 (left handed) Ṕ= 89 = 19.9= 19.9 are left handed 447 20 4.8 Measures of Association between Variables Univariate: One variable Bivariate: Two variables Observing two variables x= age of car y=annual maintenance cost of car Association (correlation) between variables (Magnitude of association, Sign -positive or negative- of associaton) *Magnitude- how big? Sign- which way?* Scatter Plots Objective: 1) construct scatter plot 2) calculate and interpret the correlation One point for each pair of data (x,y) ex) Annual Maintenance Costs of Vehicles x y 2000 2 245 1500 4 400 1000 5 450 Co500($) 8 602 0 9 799 0 5 10 15 12 1755 Age of car The scatter plot SUGGESTS a positive strong association between x&y. *Do not join the points!!* Finding a Measure of Association Correlation r= 1 x ∑(x-ẍ)-(y-ÿ) -1 ≤ r ≥ 1 -1 is a perfect negative correlation n-1 (x )(y ) 0 is no linear correlation 1 is a perfect positive correlation Correlation: a measure of LINEAR association between two variables x= independent (explanatory) variable y= dependent (response) variable

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