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# Class Note for STAT 528 at OSU 64

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Date Created: 02/06/15
Statistics 528 Data Analysis Lecture 6 July 11 2006 Christopher Holloman The Ohio spam Univers ty Summer 2006 Overview of Today s Lecture o IPS Sections 45 51 General Probability Rules Sampling Distributions for Counts and Proportions Christopher Holloman The Ohio spam Univers ty Summer 2006 General Probability Rules General Addition Rule for Union of Two Events For any two events A and B PA or B PA PB PA and B Christopher Holloman The Ohio State Univers ty Summer 2006 Christopher Holloman The Ohio State Univers ty Summer 2006 What is PA and B o If A and B are independent then by the multiplication rule PA and B PAPB 0 HA and B are dependent then we need a general multiplication rule PA and B PAPBA where PBA is the conditionalprobability that B occurs given the information that A occurs Christopher Holloman The Ohio spam Univers ty Summer 2006 o Definition of independence Two events A and B are independent if PAIB PA and PBIA PB So our old multiplication rule for independent events is a special case of the general multiplication rule o Definition of conditional probability when PA gt 0 the conditional probability of B given A is PBIA PA and B PA Christopher Holloman The Ohio spam Univers ty Summer 2006 Bayes Rule o Bayes Rule provides a formula for calculating a conditional probability of A B when only information about B A and A unconditional is available PB I APA PB I APA PB I ACPAC PAB Christopher Holloman The Ohio seam Univers ty Summer 2006 Example The OSU football stadium management purchases a new device to detect whether a person might be carrying alcohol into a game The manufacturer suggests that the device is very accurate returning a false gositive saying alcohol is present when it is not in only 1 out of 500 tests and a false negative sayin alcohol is not present when it is in 1 out of 100 tests The sta ium of cials estimate that in the future the chance of any individual attempting to carry alcohol Into a game Is 1 In 10 What Is the probabIIIty a person Is carryIng alcoho given that the machine says that they are Christopher Holloman The Ohio seam Univers ty Summer 2006 Sampling Distributions Sampling Distributions A statistic from a random sample or randomized experiment is a random variable The probability distribution of the statistic is its sampling distribution Pogulation Distributions The population distribution of a variable is the distribution of its values for all members of the population The population distribution is also the probability distribution of the variable when we choose one individual from the population at random Sampling Distribution for Counts and Proportlons Count X number of occurrences of some outcome In a fixed number of observations Sample Proportion number of occurrences out of the number of observations Xn Christopher Holloman The Ohio spam Univers ty Summer 2006 The Binomial Distribution The Binomial Setting 1 There are a fixed number n of observations 2 The n observations are all independent 3 Each of the observations falls into one ofjust two categories which for convenience we call success and failure 4 The probability of a success call it p is the same for each observation Christopher Holloman The Ohio spam Univers ty Summer 2006 Binomial Distribution The distribution of the count X of successes in the binomial setting is called the binomial distribution with parameters n and p o n is the number of observations 0 p is the probability of a success on any one observation 0 The possible values of X are whole numbers from 0 to n Notation X is Bn p or X N Bn p Christopher Holloman The Ohio spam Univers ty Summer 2006 For each of the following situations indicate whether the binomial distribution is a reasonable probability model for the random variable X a You observe the sex of the next 50 children born at a local hospital X is the number of girls among them b A couple decides to continue to have children until their first girl is born X is the total number of children the couple has Christopher Holloman The Ohio spam Univers ty Summer 2006 c A chemist repeats a solubility test 10 times on the same substance Each test is conducted at a temperature 10 degrees higher than the previous test She counts the number of times that the substance dissolves d Fifty students are tau ht about the binomial distribution in class e are the iven a short quiz on the subject The num er of stu ents who pass is counted Christopher Holloman The Ohio spam Univers ty Summer 2006 The Binomial Distribution as a Sampling Distribution We use the binomial distribution in statistical inference when we have count data in the binomial setting and we want to make inferences about the total proportion of successes in a population When the population is much larger than the sample the count X of successes in a SRS of size n has approximate a Bnp distribution if the population proportion o successes is p Rule of thumb use the binomial sampling distribution for counts when the population is at least 10 times larger than the sample Christopher Holloman The Ohio spam Univers ty Summer 2006 Finding Binomial Probabilities Assume X N Bnp if k lt n then PX k ijka p Vt n where Note a aa1a21 and 0 1 gt This is the binomial formula 16 Example Free Throws Lecture 5 X is the number of shots made in three attempts x PXx 0 008 1 01010103 2 014014014 042 3 02 Christopher Holloman The Ohio spam Univers ty Summer 2006 X N B3 058 so we can use the binomial formula to get the probability distribution of X For example PX 205821 0581 042 Christopher Holloman The Ohio spam Univers ty Summer 2006 Finding Binomial Probabilities Using Minitab Imagine that we are interested in knowing the probability that a 058 percent shooter makes 10 out 20 freethrows 1 PX 10 GSJQSSYU1 O5820 10 Christopher Holloman The Ohio spam Univers ty Summer 2006 2 Using Minitab Calc Probability Distributions Binomial Check Probability Number of Trials 20 Probability of Success 058 Check Input Constant 10 gt PX10 01359 Christopher Holloman The Ohio spam Univers ty Summer 2006 10 Using Minitab to find cumulative probabilities eg PX lt 8 PX lt 7 Instead of findin PXlt7 PX0PX gP 7 directly use the cumulative pro ability feature Calc Probability Distributions Binomial Check Cumulative Probability instead of Probability Number of Trials 20 Probability of Success 058 Check Input Constant 7 gt PXlt8 00324 Christopher Holloman The Ohio spam Univers ty Summer 2006 Binomial Mean and Standard Deviation If X N Bnp what are ux and ex X is the number of successes in n independent observations that each have probability of success p Let S be the random variable that indicates whether the itl observation was a success Si 1 or a failure Si 0 The distribution of each Si is Outcome 1 0 Probability p 1 p Christopher Holloman The Ohio spam Univers ty Summer 2006 11 Using the definitions of the mean and variance of discrete random variables us 1p 01p p 625 1p2p 0D2 1D p 1D Christopher Holloman The Ohio Sham Univers ty Summer 2006 Since X S1 52 S then Pxnp 02x POP 0quot 0x up 1 17 Christopher Holloman The Ohio Sham Univers ty Summer 2006 12 What about the sample proportion Xn Using the rule for linear functions of random variables lJ P G ip n Christopher Holloman The Ohio spam Univers ty Summer 2006 Question What does the sampling distribution of X look like Use Minitab enerate lots of samples from a B20058distribution Calc Random Data Binomial Generate 1000 rows of data Store in Columns C1 Number of Trials 20 Probability of Success 058 Look at the histogram of C1 Christopher Holloman The Ohio spam Univers ty Summer 2006 13 Question What does the sampling distribution of look like Create a new column with the proportions Calc Calculator Store result in C2 Expression C120 Look at the histogram of C2 Christopher Holloman The Ohio seam Univers ty Summer 2006 Normal Approximation for Counts and Proportions o X is approximately Nnp lnp1 p Nu039 0 is approximately Np g Nua Rule of Thumb This approximation is only valid for values of n and p that satisfy np gt 10 and n1p gt 10 Christopher Holloman The Ohio seam Univers ty Summer 2006 14

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