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# 1/11-1/15 3460

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## Popular in Statistics

## Popular in Statistics

This 5 page Class Notes was uploaded by Raghad Kodvawala on Sunday January 10, 2016. The Class Notes belongs to 3460 at Ohio State University taught by Laura Kubatko in Spring 2016. Since its upload, it has received 39 views. For similar materials see Statistics in Statistics at Ohio State University.

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Date Created: 01/10/16

Statistics the science of analyzing data o The objective of statistics is to make inferences about a population based on the information contained in a sample from that population and to provide an associated measure of goodness for 1.1: Sampling Population the entire collection of objects or outcomes about which information is sought Sample a subset of a population, containing the objects or outcomes that are actually observed o Simple random sample a sample chosen by a method in which each collection of n population items is equally likely to comprise the sample o Sample of convenience a sample that is not drawn by a well-defined random method o Sampling variation when two different samples from the same population differ from each other Parameter characteristic of the population o Example: Finite population Parameter: number of hours OSU students spend studying during finals week Population: all OSU students Sample: 100 students Statistic: average of the values for the students in the sample Statistic a value computed from the observations in a sample Tangible and conceptual populations o Tangible population a finite population consisting of actual physical objects o Conceptual population an infinite population from an experiment consisting of all values that could possibly have been observed Independence o Independent when knowing some of the items in a sample does not help predict the values of the others Occurs in large populations Values are only independent when the sample comprises 5% of the population or less o Sampling with replacement replacing each item back into the population after it is sampled Other sampling methods o Weighted sampling a sample in which some items are given a greater chance of being selected than others o Stratified random sampling the population is divided into subpopulations and a simple random sample is drawn from each Strata a subpopulation o Cluster sampling the population is divided into clusters and samples are taken from the clusters Types of data o Numerical when the results of an experiment are numbers o Qualitative when the results are descriptive observations o Categorical when sample items are placed into categories Controlled experiments and observational studies o Controlled experiment an experiment in which the values of the factors are under the control of the experimenter Produce reliable information about cause-and-effect relationships o Observational study when the experimenter simply observes the sample as they are without having any control over them 1.2: Summary Statistics 1 Sample mean X = ∑ X i n o Example: Systolic blood pressure n = 9 112, 128, 108, 129, 125, 153, 155, 132, 180 mmHg X1= 112 X 2 128 X9= 180 1 X´ = 9 (112 + 128 + … + 180) 1 2 Sample variance s = ∑ (X iX) n−1 o Example: Systolic blood pressure 1 s = 8 [(112 – 138) + (128 – 138) + … + (180 – 138) ] = 550 1 ´ 2 Sample standard deviation s = n−1 ∑ (Xi−X) √ o If X , … , X is a sample and Y = a +bX, where a and b are constants 1 n i i then Y = a + b X o If X1, … , n is a sample and Yi= a +bX,iwhere a and b are constants then sy2 = b sx2and s =y|b|s x Outliers points that are much larger or much smaller than the others in the sample o Should be scrutinized, and any resulting from error should be canceled or deleted The sample median n+1 o If n is odd the sample median, m, is the number in position 2 n o If n is even, m is the average of the numbers in positions 2 and n + 1 2 Example: Systolic blood pressure 108, 112, 125, 128, 129, 132, 153, 155, 180 m = 129 mmHg Quartiles divste the sample into quarters o Q 1 1 quartile 0.25(n+1) If this number is an integer then Q1equals the value in that position, if not 1 equals the value of the average of the two values on either side of that position o Q 2 2 ndquartile median o Q 3 3 quartile 0.75(n+1) If this number is an integer then Q3equals the value in that position, if not 3 equals the average of the two values on either side of that position Example: Systolic blood pressure nd rd o Q 1 0.25(9+1) = 2.5 average of 2 and 3 112+125 = 2 = 118.5 mmHg o Q 2 129 mmHg o Q = 0.75(9+1) = 7.5 average of 7 and 8 th 3 153+155 = 2 = 154 mmHg Descriptive statistics statistics that accurately describe the data s Standard error of the mean = √ n o Not a descriptive statistic but important for constructing confidence intervals and descriptive statistics 1.3: Graphical Summaries Stem-and-Leaf Plots o Stem the leftmost one or two digits o Leaf the rightmost digit Example: Systolic blood pressure Stem Leaf 10 8 11 2 12 589 13 2 14 15 35 16 17 18 0 Dotplot a graph that can be used to give a rough impression of the shape of a sample o For each value in the sample, a vertical column of data is drawn, with the number of dots in the column equal to the number of times the value appears in the sample Example: Systolic blood pressure • • • •• • • • • | | | | | | | | | 100 110 120 130 140 150 160 170 180 Histogram a graph that gives an idea of the “shape” of a sample o Class interval intervals that divide the sample into groups o Frequency table a table that indicates the number of points in each class interval Symmetry and skewness o Symmetric when the right half of a histogram is similar to the left half o Skewed when a histogram is not symmetric Skewed to the right when a histogram has a long right-hand tail Skewed to the left when a histogram has a long left-hand tail Unimodal and bimodal histograms o Mode the most frequently occurring value in a sample o Unimodal when a histogram only has one peak o Bimodal when a histogram has two distinct peaks Boxplots a graphic that presents the median, the first and third quartiles, and any outliers that are present in a sample o Interquartile range the difference between the first and third quartile o Extreme outlier any point that is 1.5 IQR above the third quartile or 1.5 IQR below the first quartile Comparative boxplots o Several boxplots are often placed side-by-side for visual comparison of the several samples 3.1: Basic Ideas Probability the study of randomness Experiment a process that results in an outcome that cannot be predicted in advance with certainty Sample space the set of all possible outcomes o Example: Roll a die S = {1, 2, 3, 4, 5, 6} Event a subset of a sample space o Example: Roll a die A = “outcome is even” A = {2, 4, 6} B = “outcome ≥ 4 B = {4, 5, 6} C = “outcome is odd” C = {1, 3, 5} Combining events o Union the set of outcomes that belong to A, to B, or both o Intersect the set of outcomes that belong to A and to B o Complement the set of outcomes that are not in A Example: Roll a die Union = AᴜB = {2, 4, 5, 6} Intersect = A∩B = {4, 6} Complement = A = {1, 3, 5} C B = {1, 2, 3} Mutually exclusive events two events that have no outcomes in common and so can never occur simultaneously o Example: Roll a die A and B are not mutually exclusive A∩B = {4, 6} A and C are mutually exclusive A∩C = ᴓ B and C are not mutually exclusive B∩C = {5} Probability a measure of the likelihood that an event will occur o Axioms of probability Let S be a sample space. P(S) = 1 For any event A, 0 ≤ P(A) ≤ 1 If A and B are mutually exclusive, then P(AᴜB) = P(A) + P(B) o For any event A, P(A) = 1 – P(A )C The addition rule P(AᴜB) = P(A) + P(B) – P(A∩B) o Example: Roll a die P(A) = P({2, 4, 6}) = P({2}) + P({4}) + P({6}) = ⅙ + ⅙ + ⅙ = /6= ½ C A = C = 1 – P(A) P(S) = P(AᴜC) = P(A) + P(C) = 1 P(AᴜB) = / ≠ P(A) + P(B) 3 6 3 2 2 P(AᴜB) = / +6/ – 6 = /6 3

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