Statistics 371 1 Chapter 3 5 Summary Fall 2002 Supplementary notes on probability 1 1 Random samples A simple random sample of n items is one in which each all possible subsets of size n are equally likely This can arise from drawing names from a hat one at a time where at each draw all of the remaining names in the hat are equally likely and there is no dependence on previous draws from the hat Samples that are not random are prone to bias Recall the classroom exercise where the class distribution of average rectangle size from judgment samples was centered to the right of the center of the distribution of random sample means 1 2 Probability Probability measures the likelihood of events on a scale from 0 to 1 A random variable is a variable whose value is randomly determined The probability distribution of a discrete random variable places discrete chunks of probability at specific locations The distribution can be described with a table that lists the possible values of the random variable and the probability associated with each Sometimes we use a formula to specify the probability of each possible outcome It is always the case with discrete random varaibles that the sum the probabilities of each possible outcome is one Probability distributions of continuous random variables describes how thick a probability dust is spread over the line A probability density curve is a nonnegative function where the total area under the curve is one that has the property that the area under the curve between two points a and b is the probability that the random variable is between a and b 1 3 The binomial distribution The binomial distribution arises from counting the number of heads in a prespecified number of coin tosses This is a model for the way that data is produced for a vast number of examples in statistics In particular we will use this model when examining the proportion of a random sample that belongs to a particular category Every binomial random variable is described by two parameters n is the number of trials and p is the probability that an individual trial is a success The binomial setting You may recognize a setting in which the binomial distribution is appropriate with the acronym BINS binary outcomes independent trials n is fixed in advance same value of p for all trials A trial has one of two possible values One is called a success and the other is called a failure We want to count the number of successes The binomial distribution is appropriate when we have this setting 1 there are a fixed number of trials 2 there are two possible outcomes for each trial 3 the trials are independent of one another 4 there is the same chance of success for each trial 5 we count the number of successes The binomial probability formula for exactly j successes and n j failures in n independent trials with success probability p is n Pr Y j n Cj pj 1 p n j for j 0 1 n where n Cj j n j There is no simple formula to sum binomial probabilities to calculate the probability that a binomial random variable is one of several outcomes you need to compute the outcomes individually and sum them Bret Larget October 2 2002 Statistics 371 1 4 Chapter 3 5 Summary Fall 2002 The normal distribution Many naturally occuring variables have distributions that are well approximated by a bell shaped curve or a normal distribution These variables have histograms which are approximately symmetric have a single mode in the center and tail away to both sides Two parameters the mean and the standard deviation describe a normal distribution completely and allow one to approximate the proportions of observations in any interval by finding corresponding areas under the appropriate normal curve In addition the sampling distributions of important statistics such as the sample mean are approximately normal for moderately large samples for many populations Characteristics of all normal curves Each bell shaped normal curve is symmetric and centered at its mean The total area under the curve is 1 About 68 of the area is within one standard deviation of the mean about 95 of the area is within two standard deviations of the mean and almost all 99 7 of the area is within three standard deviations of the mean The places where the normal curve is steepest are a standard deviation below and above the mean and Standardization In working with normal curves the first step in a calculation is invariably to standardize z x This z score tells how many standard deviations an observation x is from the mean Positive z scores are greater than the mean and negative z scores are below the mean If the z score is known and the value of x is needed solving the previous equation for x gives x z Reading the algebra this simply states that x is z standard deviations above the mean The standard normal distribution Areas under all normal curves are related For example the area to the right of 1 76 standard deviations above the mean is identical for all normal curves Because of this we can find an area over an interval for any normal curve by finding the corresponding area under a standard normal curve which has mean 0 and standard deviation 1 Using the normal table The standard normal table is located in the inside cover of your textbook It tells you the area to the left of z Because the normal curve is symmetric and the total area under the curve is 1 this is sufficient to find the area under the curve over any interval You will need to be able to use the table to find areas when the numbers on the axis are known and to be able to use the table to find numbers on the axis when areas are known It is helpful to draw a sketch of a normal curve in working out problems Draw one axis with the units of the problem Draw a second axis with standard units The normal approximation to the binomial distribution This is useful background information but we will not test it You have access to software to compute sums of binomial probabilities exactly so there is no need to use a normal approximation except as a check Consider this example find the probability that there are 410 or more successes in 500 independent trials when the probability of success on a single trial is 0 8 An exact expression of this probability is P X 410 500 X 500 0 75 x 0 25 500 x x 500 x x 410 Even with a calculator this is an intimidating computation Statistical software gives the answer numerically as 0 1437 Bret Larget October 2 2002 Statistics 371 Chapter 3 5 Summary Fall 2002 When the
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