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Midterm

# 2320 Formula Sheet Midterm.doc

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• Descriptive Statistics are methods of organizing, summarizing, and presenting data in a convenient and - Second (middle) quartile,Q2,= 50th percentile - Ninth (upper) decile = 90th percentile informative way. These methods include: Graphical Techniques and Numerical Techniques. - Third quartile, Q3, = 75th percentile • Inferential Statistics is set of methods used to draw conclusions or inferences about characteristics ofnterquartile range = 3 – Q1 (measure the spread of the middle 50% of observation) populations based on data from a sample. Location of a Percentil, L P (n+1) (P/100) where Lpis the location of the Pth percentile E.g. {0 1 2 3 4 5 6 7 759} L = (10+1) (75/100) = 8.25  The third quartiles or 75 percentile = 7 + (8-7) 0.25 = 7.25 • Statistical Inference is the process of making an estimate, prediction, or decision about a populatioMeasures of Linear Relationship - strength and direction of a linear relationship between two variables based on a sample. Conclusions and estimates are not always going to be correct, so we build into statistical inference a measure of reliability: the confidence level and the significance level. Popular: • Confidence Level (1-α) is the proportion of times that an estimating procedure will be correct. (x − μ )(y − μ ) E.g. x is hours2of s2udy and y is grade ∑ i x i y n x y x y xy • Significance Level (α) measures how frequently the conclusion will be wrong in the long run. σ xy = 1 0 40 0 1600 0 • E.g. the poll is considered accurate within 3.4 percentage points, 19 times out of 20. Our Confidential N 3 8 65 64 4225 520 level is 95% (19/20 =0.95), while our significance level is 5%. Sample: 4 10 70 100 4900 700 5 15 85 225 7225 1275 • There are three types of data: Interval, Nominal, and Ordinal data. Covariance ∑ (x i x)(y − y)i 6 20 100 400 10000 2000 sxy = 58 415 814 30975 4770 • Data  Not Categorical  Interval ( real number; all calculation; treated as ordinal or nominal) n −1  Categorical  Order  Ordinal (ranked order; Calculation only based on ordering process; Shortcut for Sample Covariance: s = 1 4770 − 58 × 415  == 151.6667 May be treated as nominal but not interval) xy   1  ∑ ∑ x i y i 6 −1  6   No Order  Nominal (arbitrary numbers represent categories, Only calculate based on sxy =  ∑ x i i  2 frequency; May not be treated as nominal or not interval ) n − 1  n  1  (58)  • Single set of Nominal Data:Use Frequency distribution and Relative Frequency Tables, Bar and Pie s x 814 −  = 50.6667 • Histogram is most important graphical method for single set of Interval Data; it doesn’t only summarize σ 6 −1  6  interval data but also help explain probabilities. Popular: ρ = xy 2 σ σ 2 1  (415)  o Number of classes intervals = 1 + 3.3 log (n) x y s y 30975 −  = 454.1667 o Class width = (Largest observation + Smallest observation) / Number of Classes. − 1 ≤ ρ ≤ +1 6 −1  6  • Cross-Sectional Data  Observations measured at the same point in time. s s = s2 = 50 .6667 = 7.1181 • Time-Series Data  Observations measured at successive points in time  Line chart Coefficent of Sample: r = xy x x • Skewness: A skewed histogram is one with a long tail extending to either the right or the left. Correlation s s −1≤ r ≤ +1 2 Modality: A unimodal histogram is one with a single peak, while a bimodal histogram has two peaks. x y s y sy = 454 .1667 = 21 .3112 • Bell Shape: A special of symmetric unimodal histogram. r = + 1 Strong positive linear relationship r = 0 No linear relationship sxy 151 .6667 • Ogive is a graph of a cumulative relative frequency distriWe use Ogive to answer: about 48% of the r = – 1 Strong negative linear relationship = = = 0.9998 students have first income lower than \$3000. ….85% lower than \$4500 etc. r = 0.56 Moderately strong positive sxs y 7.1181 × 21.3112 • Two Nominal Variable  Contingency Table and Bar Chart (two-dimensional) r = – 0.1 Weak negative ….. • Two Interval Variable  Scatter diagram to explore the relationship between 2 interval data. Independent variable ( X )  Horizontal axis. Dependent variable ( Y )  Vertical axis. • Marginal Probabilities are computed by adding across rows and down columns. B 1 B2 P(Ai • Linearity (Linear Rela.) Scatter diagram  Positive or Negative, strong or weak or non-linear relations. A2 .20 .35 .55 Marginal If a distribution is symmetrical, If the distribution is nonsymmetrical, and skewed Joint A2 .15 .30 .45 Probabilities the mean, median and mod may coincide to the left or the right, the three measure differ. Probabilities P(Bi .35 .65 1.00 • A Random Experiment is an action or process that leads to one of several possible outcomes. The listed outcomes must be exhaustive (All possible outcomes included), and the outcomes must be mutually exclusive (No two outcomes can occur at the same time.) • Sample Space of a random experiment is a list of exhaustive and mutually exclusive outcomes. The probability of any outcome is between 0 and 1, and the sum of the probabilities of all the outcomes equals 1. S = {O , O , …, O } 0 ≤ P(O) ≤ 1 1 2 k i ∑ P (O )i= 1 k Interval Chebysheff Theorem 1 − 1 Empirical Rule (Bell shape) • There are three ways to assign a probability to an outcome: Classical Approach (Fair chance), Relative 2 Frequency (experimentation or historical), and Subjective Approach (the assignor’s judgment). k • An Event is a collection or set of one or more simple events in a sample space. A Simple Event is 1 x − 1s, x + 1s At least 0% (1 – 1/1 ) = 0 Approximately 68% individual outcome of a sample space. 2 x − 2s, x + 2s At least 75% (1 – 1/2 ) = 0.75 Approximately 95% • The Probability of an Event is the sum of the probabilities of the simple events that constitute the event. 2 • The Complement of Event A is defined to be the event consisting of all sample points that are “not in A” 3 x − 3s, x + 3s At least 88.9% (1 – 1/3 ) = 0.889 Approximately 99.7% or A . P(A) + P(A ) = 1 • Measures of Relative Standing are designed to provide information about the position of parti
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