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Stats 121 notes week 6

by: Sydney Clark

Stats 121 notes week 6 STAT 121

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Sydney Clark
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Discussing probability and correct experiments to successfully collect CORRECT data.
Principles of Statistics
Dr. Christopher Reese
Class Notes
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This 5 page Class Notes was uploaded by Sydney Clark on Friday October 7, 2016. The Class Notes belongs to STAT 121 at Brigham Young University taught by Dr. Christopher Reese in Winter 2016. Since its upload, it has received 2 views.

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Date Created: 10/07/16
Stats 121 Notes Week 6 *IMPORTANT*  Randomized Controlled Experiment: subjects randomly assigned to treatments  o Use when individuals are similar   Randomized Block Design (RBD)  o matched pairs, a special case of RBD o What is a block?  A group of individuals that are:  Similar with respect to some characteristics known before the  experiment begins, and that characteristic is expected to affect the  response to the treatments  Often equal in number to the number of treatments.  Within blocks the subjects are very homogenous and outside of  blocks the subjects are very different o What is a randomized block design?  An experimental design where the random assignment of individuals to  treatments is carried out separately within each block. o NOTE: blocks are another form of control  they control the effects of the variables that define the blocks. o Make design more powerful by perfectly equalizing effects of certain lurking  variables   1. classify subjects into blocks based on lurking variable(s)   2. randomly assign subjects to treatments separately within each block o The number of subjects in a block has to be equal to the number of treatments  Example:  o Use when the individuals are similar within a block but very different from block  to block  RBD removes confounding of lurking variables with response variable    RBD reduces chance variation by removing variation associated with the  lurking (blocking) variable.    RBD yields more precise estimates of chance variation which makes  detection of statistical significance easier 9 / 16 Matched Pairs  o Matched pairs setting  Special case of randomized block designs   Block: Pair of individuals or pair of measurements   Explanatory variable: two treatments   Examples:   o Twins: each receiving a treatment  o  Two treatments on each individual  o Measurements before and after treatment on each  individual o Three principles of experiments:  1.) Randomly assign the 2 treatments to the two individuals within each  pair (block) OR randomize the order of applying the treatment to each  individual  2.) Replication equals the number of pairs  3.) Compare the 2 treatments. Each pair serves as its own control  Analogy o Taking a simple random sample from a population and calculating a summary  statistic analogous to game of chance:   1. perform random procedure with many possible outcomes   2. end up with one particular outcome   3. distribution of outcomes for large number of plays can be characterized  Example:  o 1. roll a fair die; numbers 1­6 are possible outcomes  o 2. we get, say, 4  o 3. if die is fair and rolled many times, 1/6  of rolls should  give 1, 1/6 th should give 2, etc.  Probability theory o Components  1.Specify game (including strategy)  2.Specify possible outcomes (the sample space)  3.specify probability distribution=long run of proportions associated with  each possible outcome o Theory can guide decision on strategy for playing game:   strategy that has higher probability (long­term proportion) of favorable  results can be considered better strategy (even in short run)  Understand how to reproduce experiment o Playing the game over and over again o Observing the frequency of winning  Random Phenomenon o individual outcome unpredictable, but outcomes from large number of repetitions  follow regular pattern  Sample space o Set of all possible outcomes  Event o a collection of possible outcomes   Example: We can write the event “rolling an odd number on a die” as the  set {1, 3, 5}  Probability outcome o The proportion of times that an outcome occurs in many, many repetitions (plays)  of the random phenomenon o PR(x=1)  Probability rules o Must be between 0 and 1 o Sum of probabilities from all possible outcomes must equal 1 o If 2 events cannot occur simultaneously, the probability either one or the other  occurs equals the sum of their probabilities  Mutually exclusive o The probability that an event does not occur equals 1 minus the probability that  the event does occur  Probability of the compliment  If we let “A” be an event, the probability of A (“P(A)”) is the of that event happening  o  For any event A, P(A) is always in the interval [0, 1]   X=1 o P(A)=0 means A will certainly NOT happen  o P(A)=0.5 means the chance of A occurring is equal to the chance A does NOT  occur o P(A)=1 means A certainly WILL occur  LIST OUT ALL POSSIBLE OUTCOMES  Empirical probability o Empirical (or observational or long­run) probability – approximated by playing  the game (running the experiment) many times and observing frequency of  occurrence    Example: P(“instructor makes a 15­foot shot with basketball”) ≈ 0.65,  because she made 13 out of 20    Empirical probabilities are approximate, but theoretical probabilities are  often impossible to calculate; e.g., P(“randomly choosing a student that is  male, a stat major, and from California”)  Do the experiment over and over and over again  An empirical probability is obtained from the relative frequency of the  event, where   Law of Large Numbers: As the number of trials (or repetitions) of the  experiment/game increases, the relative frequency of the event gets closer  and closer to the theoretical probability of the event  Random variable  Ex: Major in college o Can take on random values o The mapping of a name to a number  Probability distribution o Set possible outcomes in a sample space AND the % associated with each  outcome  Ex: all presidential candidates represented from people in salt lake county o Probabilities must sum to 1 o Can be represented by table, formula, or graph  Random variable o characteristic that is measured on each individual; e.g., cost, height, yield, gender  Continuous Random variable o variable that can take on any value in an interval so that all possible values cannot be listed; e.g., time, height, temperature  Discrete random variable o variable whose possible values are a list of distinct values; e.g., gender, opinion, # of arrests, shoe size.  Two types of discrete random variable:  o  Discrete categorical (e.g., college major)  o  Discrete quantitative (e.g., persons living in household)  Interpretations  o DON’T: Try to describe the shape or calculate measures of center or spread (e.g.,  mean, median, standard deviation, IQR) o DO: compare percentages for outcomes o Compare percentages  o Calculate appropriate measures of center or spread (e.g., mean, median, standard  deviation, IQR)  Continuous Random variable o Can take on any value within the range of the variable  o We focus on the probability that a value is in a specific interval  Ex: Probability that a person’s height is between 67.5 inches and 68.5  inches  Note: With a precise enough ruler  Interpretations o Compare percentages  o  Calculate appropriate measures of center or spread  It often makes more sense to model the probability distribution with a smooth curve  called the probability density curve 2  Smooth curve is a model (Probability modeling curve) o curve is on or above horizontal line (x­axis) o  area under the curve = 1  o where curve is high, data values are dense  o does not describe distribution exactly – accurate enough for practical purposes   often gives more accurate estimates of probabilities than using the  histogram of your sample data o We calculate probability of a random variable taking on values of an interval that  is equal to the area under the curve for that interval


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