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BBA182 Applied Statistics Week 5 (1) Empirical rule - Probabilities Dr Susanne Hansen Saral Email: susanne.saral@okan.edu.tr

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Interpretation of summary statistics A random sample of people attended a recent soccer match. The summary statistics (Excel output) about their ages is here below: What is the sample size? What is the mean age? What is the median? What shape does the distribution of ages have? (symmetric or non-symmetric) What is the measure/s for spread in the data? Is this a large spread? What is the Coefficient of variation for this data?

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Deviations from the normal distribution - Kurtosis A distribution with positive kurtosis is pointy and a distribution with a negative kurtosis is flatter than a normal distribution

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Positively and negatively skewed Positive skewed is when the distribution is skewed to the right Negative skewed is when the distribution is skewed to the left

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Symmetric distribution - Empirical rule Knowing the mean and the standard deviation of a data set we can extract a lot of information about the location of our data. The information depends on the shape of the histogram (symmetric, skewed, etc.). If the histogram is symmetric or bell-shaped, we can use the Empirical rule.

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Probability as Area Under the Curve

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If the data distribution is symmetric/normal, then the interval: If the data distribution is symmetric/normal, then the interval: contains about 68% of the values in the population or the sample

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contains about 95% of the values in the population or the sample contains almost all (about 99.7%) of the values in the population or the sample

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Empirical rule: Application A company produces batteries with a mean lifetime of 1’200 hours and a standard deviation of 50 hours. Find the interval for (what values fall into the following interval?):

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Empirical rule: Application A company produces batteries with a mean lifetime of 1’200 hours and a standard deviation of 50 hours. The mean is 1’200 and standard deviation is 50, we find the following intervals: = 1’200 1x(50) = (1’150 and 1’250) = 1’200 2x(50) = (1’100 and 1’300) = 1’200 3x(50) = (1’050 and 1’350)

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Interpretation of the Empirical rule: Lightbulb lifetime example If the shape of the distribution is normal, then we can conclude : That approximately 68% of the batteries will last between 1’150 and 1’250 hours That approximately 95% of the batteries will last between 1’100 and 1’300 hours and That 99.7% (almost all batteries) will last between 1’050 and 1’350 hours. It would be very unusual for a battery to loose it’s energy in ex. 600 hours or 1’600 hours. Such values are possible, but not very likely. Their lifetimes would be considered to be outliers

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Empirical rule exercise If the mean, of a population is 250 and its standard deviation, , is 20, approximately what percent of observations is in the interval between each pair of values? A. 190 and 310 B. 210 and 290 C. 230 and 270

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Class quizz Empirical rule: (1) Which shape must the distribution have to be able to apply the Empirical rule? (2) Which two parameters give information about the shape of a distribution? (3) What percent approximately of the values in a normal distribution are within one standard deviation above and below the mean ?

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Introduction to Probabilities

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Probability theory “Life would be simpler if we knew for certain what was going to happen in the future” B. Render, R. Stair, Jr. M. Hanna & T. Hale, Quantitative Analysis for Management, 2015 However, risk and uncertainty is a part of our lives

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Definition of probability Probability is a numerical measure about the likelihood that an event will occur.

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Probability and time Time Certainty Uncertainty Certainty runs over a short period of time and gradually decreases as the time horizon becomes more distant and uncertain.

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Probability and its measures: 2 basic rules Rule 1: Probability is measured over a range from 1 to 0 ( 0 – 100%) Probability – the chance that an uncertain event will occur (tossing a coin)

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Probability and its measures 2 basic rules Rule 2: The sum of the simple probabilities for all possible outcomes of an activity must equal 1. P(x) = = 1 Regardless how probabilities are determined, they must stick to these two rules

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Probability rule 1 and 2 applied - example

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Probability and definitions Random experiment Sample space Sample point Event

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Random experiment In statistics a random experiment is a process that generates two or more possible, well defined outcomes. However, we do not know which of the outcomes will occur next. Examples: Experimental outcomes: Tossing a coin Head, tail Throwing a die 1, 2, 3, 4, 5, 6 The outcome of a football match win – lose - equalize – game cancelled

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All possible experimental outcomes constitute the sample space A sample space (S) of an experiment is a list of all possible outcomes. The outcomes must be collectively exhaustive and mutually exclusive.

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Sample space, S - Examples Random experiment: Flip a coin Possible outcomes: Head or tail The sample space: S= {head, tail} There are no other possible outcomes, therefore they are collectively exhaustive. When head occurs, tail cannot occur – meaning the outcomes are mutually exclusive. The sample points in this example are head and tail.

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Sample space, S - Examples Outcomes of a statistics course: The sample space: S = {AA, BA, BB, CB, CC, DC, DD, FD, FF, VF)}. There are no other possible outcomes, therefore they are collectively exhaustive. When one of the outcomes occur, no other outcome can occur, therefore they are mutually exclusive. The sample points are the individual outcomes of the sample space, S = {AA, BA, BB, CB, CC, DC, DD, FD, FF, VF}.

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Sample space - example The sample space, S = { Google, direct, Yahoo, MSN and all other} Mutually exclusive: When a person visits Google it can not visit Yahoo at the same time Collectively exhaustive: There are no other possible search engines Sample points: Google, Direct, Yahoo, MSN, all others

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Event An individual outcome of a sample space is called a simple event. An event is a collection or set of one or more simple events in a sample space.

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Event: – subset of outcomes of a sample space, S Random experiment: Throw a dice (Turkish: zar). Possible outcomes, sample space, S is: {1, 2, 3, 4, 5, 6} We can define the event “toss only even numbers”. Let A be the event «toss only even numbers»: We use the letter A to denote the event: A: {2, 4, 6} If the experimental outcome are 2, 4, or 6, we would say that the event A has occurred.

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Event : Subset of outcomes of a sample space, S Random experiment: Grade marks on an exam Possible outcomes (Sample space): Numbers between 0 and 100 We can define an event, «achieve an A», as the set of numbers that lie between 80 and 100. Let A be the event «achieve an A»: A = (80, 81, 82 …….98, 99,100) If the outcome is a number between 80 and 100, we would say that the event A has occurred.

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Events Intersection of Events – If A and B are two events in a sample space S, then the intersection, A ∩ B, is the set of all outcomes in S that belong to both A and B. We also call this a joint event. They are not mutually exclusive since they have values in common

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Union of events Union of Events – If A and B are two events in a sample space S, then the union, A U B, is the set of all outcomes in S that belong to at least one of the two events. Therefore the union of A U B occurs if and only if either A or B or both occur.

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Mutually exclusive event A and B are Mutually Exclusive Events if they have no basic outcomes in common i.e., the set A ∩ B is empty, indicating that A ∩ B have no values in common Example: Tossing a coin: A is the event of tossing a head. B is the event of tossing a tail. They cannot occur at the same time.

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Collectively Exhaustive Events E1, E2, …,Ek are Collectively Exhaustive events if E1 U E2 U….. Ek = S i.e., the events completely cover the sample space Example: Tossing a coin - possible events: head and tail Events, head and tail are collectively exhaustive because they make up the entire sample space, S

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Complement The Complement of an event A is the set of all basic outcomes in the sample space that do not belong to A. The complement is denoted Example: Roll a die A = (all possible even numbers) = (all possible uneven numbers

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Examples – rolling a dice

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Examples Mutually exclusive: A and B are not mutually exclusive The outcomes 4 and 6 are common to both Collectively exhaustive: A and B are not collectively exhaustive A U B does not contain 1 or 3

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Class exercise The following sample space is defined a S = {2, 3, 15, 17, 21} 1) Given the event A = {3, 17}, define event 2) Given the events from the sample space, S: A = {3, 17 } and B = {3, 15, 21} Define What is the intersection of A B ? What is the union of A Are events A collectively exhaustive? Explain Are events A and B mutually exclusive? Explain

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Class exercise The following sample space is defined a S = {2, 3, 15, 17, 21} 1) Given the event A = {3, 17}, define event = {2, 15, 21} 2) Given the events from the sample space, S: A = {3, 17 } and B = {3, 15, 21} - Define = {2, 17} - What is the intersection of A B ? = {3} - What is the union of A = {3, 15, 17, 21} - Are events A collectively exhaustive? They are not collectively exhaustive because the sample point 2 is not in the union. - Are events A and B mutually exclusive? No, because event A and B have a sample point in common, 3