Statistics 371 Chapter 3–5 Summary Fall 20021 Supplementary notes on probability1.1 Random samplesA simple random sample of n items is one in which each all possible subsets of size n are equally likely. This can arise fromdrawing names from a hat one at a time, where at each draw all of the remaining names in the hat are equally likely andthere 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 averagerectangle size from judgment samples was centered to the right of the center of the distribution of random sample means.1.2 ProbabilityProbability 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 probabilityassociated with each. Sometimes, we use a formula to specify the probability of each possible outcome. It is always the casewith 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 thatthe 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 distributionThe binomial distribution arises from counting the number of heads in a prespecified number of coin tosses. This is a modelfor the way that data is produced for a vast number of examples in statistics. In particular, we will use this model whenexamining 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 thatan individual trial is a success.The binomial setting: You may recognize a setting in which the binomial distribution is appropriate with the acronymBINS: 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 thenumber 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 successesThe binomial probability formula for exactly j successes (and n−j failures) in n independent trials with success probabilityp isPr{Y = j} =nCjpj(1 −p)n−jfor j = 0, 1, . . . , n wherenCj=n!j!(n − j)!There is no simple formula to sum binomial probabilities: to calculate the probability that a binomial random variable is oneof several outcomes, you need to compute the outcomes individually and sum them.Bret Larget October 2, 2002Statistics 371 Chapter 3–5 Summary Fall 20021.4 The normal distributionMany naturally occuring variables have distributions that are well-approximated by a ”bell-shaped curve”, or a normaldistribution. These variables have histograms which are approximately symmetric, have a single mode in the center, and tailaway to both sides. Two parameters, the mean µ and the standard deviation σ describe a normal distribution completely andallow one to approximate the proportions of observations in any interval by finding corresponding areas under the appropriatenormal curve.In addition, the sampling distributions of important statistics such as the sample mean are approximately normal formoderately large samples for many populations.Characteristics of all normal curves:• Each bell-shaped normal curve is symmetric and centered at its mean µ.• 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 standarddeviations 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 (µ −σ and µ + σ).Standardization: In working with normal curves, the first step in a calculation is invariably to standardize.z =x − µσThis z-score tells how many standard deviations an observation x is from the mean. Positive z-scores are greater than themean, 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 givesx = µ + z × σReading 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 of1.76 standard deviations above the mean is identical for all normal curves. Because of this, we can find an area over aninterval for any normal curve by finding the corresponding area under a standard normal curve which has mean µ = 0 andstandard deviation σ = 1.Using the normal table: The standard normal table is located in the inside cover of your textbook. It tells you the areato the left of z. Because the normal curve is symmetric and the total area under the curve is 1, this is sufficient to find thearea 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 usethe 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 nottest it. You have access to software to compute sums of binomial probabilities exactly, so there is no need to use a normalapproximation except as a check.)Consider this example: find the probability that there are 410 or more successes in 500 independent trials when theprobability of success on a single trial is 0.8. An exact expression of this probability isP (X ≥ 410) =500Xx=410500!x!(500 − x)!(0.75)x(0.25)500−xEven with a calculator, this is
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