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# Biostatistics MATH225

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This 8 page Study Guide was uploaded by Alanna M on Monday September 19, 2016. The Study Guide belongs to MATH225 at Duquesne University taught by Dr. Goodburn in Fall 2016. Since its upload, it has received 6 views. For similar materials see Biostatistics in Mathematics at Duquesne University.

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Date Created: 09/19/16

Biostatistics Exam #1 September 23, 2016 Dr. Goodburn, Section 4 __ = term __ = formula Chapter 1: Biostatistics: a field of study that involves collecting, organizing, summarizing, and analyzing data. The making of inferences about a group based on data from a portion of that group. We study populations to get information about values of parameters (or population parameter) Parameter describes a characteristic of the population Denoted by Greek Letters (describing entire group) Generally unknown Parameters of interest: average age, average height, average weight, proportion of males/females Sample data: reflective of that population. Use sample data down the road to estimate parameters How? By calculating sample statistics. We can estimate… 1. Population mean age by calculating the sample mean age 2. Population proportions are estimated by calculating the sample proportion Sample statistics are not denoted by greek letters but by “regular” letters Sample mean: x (with line on top) = x-bar Sample proportion = p-hat Data: a collection of values for different variables of interest Variables vary based off what is being studied Data Often collected for the study. Two kinds: 1. Observational study: not influencing any contact or variables Collecting data and seeing what it tells you Taking data as it exists, don’t influence any values 2. Experiments; attempt to influence variables; a “treatment” introduced to see how, if at all, it impacts a variable(s) of the study Impacting their daily routine, behavior Treatment could be medication and could be many treatments Variable types Qualitative: not really a number or measurement. Typically a characteristic or trait (hair color or major) Quantitative: generally a measurement or number Discrete: values that don’t make sense are possible, gaps in the data, generally counts (number of parking spots in a parking garage) Continuous: no breaks in the value, any value is possible (theoretically); the value must make sense with your variable. Measurement scales 1. Nominal: value is a label; numbers are being used to represent different values of a qualitative variable Number is being used as a label to save time. Cannot calculate 2. Ordinal: (sometimes for qualitative) a nominal variable where there is a logical ordering scheme for the values Arbitrary values must be set; once design is made it falls into place Great for customer service surveys, questionnaires i.e. 1 – very effective 2 – somewhat effective 3 – not at all can calculate a mean differences, however, between these values have no meaning and tell us nothing. 3. Interval: (kind of exists in theory) quantitative variable where there is no “natural 0” and differences between values have meaning. Natural zero: zero is a possible value but it does not mean an absence of quantity Temperature really is one of the best examples for the interval scale. 0 is a value but doesn’t mean theres no temperature. 4. Ratio: An interval variable where zero means an absence of quantity, an absence of a value. Height and Weight How to sample Goal is that we want our sample data to reflect the population 1. Simple Random Sample (SRS) of size “n”: each member of the population had the same chance of being selected for the sample and every sample of size n had the same chance of being selected. How? Identify every member of the population - Give every member of the population a unique identification number that has the same number of digits. - Use a table of random digits to select the n members of the sample Random digit table 2. Simple Random Sample (SRS) Ideally the best - Give each member a unique identification number that has the same number of digits - Random digit table helps us select the n subjects for sample - Need to be same number of digits [can add 0’s in front of single digits] Ways to sample 1. SRS of size n 2. Systematic sample: randomly select the first individual, then select every K member th 3. Stratified sample: divide population you’re working with into groups on interest, then randomly select from each group. Practice problems: Pg. 13 # 1, 2, 3 pg. 17 # 1-8 Chapter 2: Descriptive Statistics Summarizing data with Frequency table Frequency: table that gives the counts for each value of the variable Frequencies add up to total Relative frequency is sometimes added to the table ???????????????????????????????????? ????????????????. ???????????????????????????????? ???????????????????????????????????? = = ???????????????????????? ???????????????? ???? Within the table there are classes. Classes are the number of rows in the tables To calculate the number of classes, we use Sturge’s Rule. Sturge’s Rule is the guideline to figure out how many rows you should have ???? = 1 + 3.22 × ???????????? ???? ▯▯ k = number of classes n = sample size Class Width must be calculated so classes span the same width. ???? ℎ????????ℎ − ???????????? ???????????????????? ????????????????ℎ = ???? = # ???????? ???????????????????????????? *Generally, round the class width up. Could round down if it captures highest value. Could turn table into graph or pie charts. Graphically, can be represented as a histogram, frequency polygon, or stemplot. Histograms Histograms can represent frequency or relative frequency. Classes are represented on the x axis while frequencies are represented on the y axis. Frequency polygons use the midpoints of the classes and frequencies. ???? + ???? ???????????? ???????????????????? = 2 The mid point for each class is used as the point. Stemplot (Stem and leaf display) - Good for small data sets - Organizes data from low to high How to make a stemplot: - Divide each value into 2 parts - Identify low and high values - Set up table with stems and leaves Numerical Summary Measures - Measures of central tendency - Measures of dispersion Mean - Average value - Add up all values, divide by # you have ▯▯▯ ???????? ???????????????????????????????????????? ???????????????? = µ = ▯ ???????????? N vs. n ???????????????????????? ???????????????? = ???? = ▯▯▯ ???? Median - Middle observation in ordered data - Divides data in half Mode - Most frequently occurring data value(s) Measures of “location” Median 1. Odd numbers – exact middle exits of data points ???????????????????????? ???????? ???????????? ???????????????????????????????? ???????????????? ???????????????? = ???? + 1 ???????? ???? + 1 2 2 2. Even number – no exact middle exists, create middle value ▯ ▯ ▯ ▯ ???????????????????????????????????? ????▯ ▯hen the median is average of values in location ???????? ▯ and ▯he next location. Median is resistant mediator because it is not affected by extreme values. Quartiles - Divides data into 4 parts - Q 1 Q 2 and Q 3 - Q is the same as the median (divides data in half) 2 - Q 1is the median of the lower half of data - Calculated same way median is - Q is the median of the upper half of data 3 - Calculated same way median is 5 Number Summary Minimum value Q 1 Median Q 3 Maximum value Interquartile Range ???????????? = ???? − ???? ▯ ▯ Used to help identify outliers Box and Whisker Plot (box plot) Graph of 5 number summary Box contains Q , 1 , 2nd Q 3 Whisker contains outliers and high and low value Outliers Low end: Any value below ???? − 1▯5 × ???????????? High end: Any value above ???? + 1▯5 × ???????????? Always check outliers. Used to either check mistakes or be aware of data Represented by dot on box and whisker plot Not connected to minimum and high valued lines Standard Deviation Measurement used to represent dispersion Attempts to calculate average distance between the data points and the mean. ▯ (???? − ????) ▯ ???????????????????????????????????????? ???????????????????????????????? ???????????????????????????????????? = ???? = ▯ ▯▯▯ ▯ ???? For sample standard deviation, replace N with n Using standard deviation… ???????????????????????????????? = ???????????????????????????????? ???????????????????????????????????? ▯ Variance is standard deviation and its units are squared. Distribution Types Symmetric distribution - From looking at the middle of a graph, two sides of graph are mirror images. - Doesn’t have to be exact mirroring Skewed right: right side of distribution extends longer than the left side Positive Skewed left: left side extends longer than right side Negative Coefficient of Variation (C.V.) Used to measure dispersion A measure of relative variation ???? ????.????.???????? ???? ???????????????????????????????????????? = ×100% ???? ???? ????.????.???????? ???????????????????????? = ×100% ???? Practice problems: Pg. 57 # 1-7, 12-16,18,19, 22,27 Chapter 3: Probability Probability is how likely something (generally an “event”) is going to happen. Notation: P(E) standing for event E happening Basic Probability Theory For an event, E, 0≤P(E)≤1 Null event: P(E) = 0 Certain event: P(E) = 1 Usually shown in decimals but could be fractions or percentages Probability Definitons Classical (priority): theoretical notion: If an event can occur N mutually exclusive and equally likely ways, and if m of those possesses trait E, then ???? ???? = ???? ???? Mutually exclusive: disjoint Outcomes that can’t happen at the same time. Each outcome is different. Relative Frequency Data based. States: If some process is repeated a large number of times, n, and if some resulting event with characteristic E occurs m times, then: ???? ???? ???? = ???? subjective: personal opinion based or value judgment. Values are not clear, can be based off feeling/opinion. Bayesian Methods Example of subjective probability. Taken in consideration the degree of belief in chance. Make use of… Prior probabilities: based on prior knowledge, experience, or data Posteriror probabilities: obtained by new info. Updating a prior probability Elementary probability properties Given a process with n mutually exclusive outcomes: E , E , 1 E2 probnbility of even E is non i regulatory number: P(E)≥0 i P(E 1 + P(E 2 + … P(E n = 1 P(E 1or E2) = P(E1 ∪ E 2 = P(E1) + P(E2) Sample space: set of all possible outcomes Complement of an Event Everything not a part of an original event E1’s complement is denoted as ???? ▯ ???? ???? ▯ + ???? ???? ▯ = 1 Joint Probability Probability that a randomly selected object possesses two (or more) characteristics at the same time Involving “and” P(E1 and E2) = P(E1 ∩ E2) Adding probabilities together General Addition Rule No overlap Add individual probabilities, subtracting join probability ???? ???? ▯ ???? ▯ = ???? ???? ▯ + ???? ???? ▯ − ????(???? ∩▯???? ) ▯ Conditional Probability When a condition is satisfied, narrowing the focus ???? ???? ▯ P▯obability of E whe1 E has 2een satisfied ???? ???? ∩ ???? ???? ???? ???? = ▯ ▯ ????(???? ) ≠ 0 ▯ ▯ ???? ???? ▯ ▯ General Multiplication Rule Useful when selecting one ???? ???? ▯ ???? ▯ = ???? ???? ▯ ????(???? |???? )▯ ▯ Independent Events If A & B are independent then the occurrence of A does not change the probability of B, and the occurrence of B does not change the probability of A. P(A|B) = P(B) P(B|A) = P(A) Above manifests when we select multiple (i.e. randomly select 3 people, what is probability all 3 are female? Multiplication rule for independent events: P(E1∩ E2) = P(E1) * P(E2) Can be extended to any number of events provided they are all independent Usually comes into play when making multiple selections

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