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Statistical Sciences
Statistical Sciences 2244A/B
Jennifer Waugh

5.1 Overview ● Arandom variable is a variable having a single numerical value, determined by chance, for each outcome of some procedure ● Aprobability distribution for a discrete random variable describes the probability for each value of the random variable ● Adiscrete random variable has either a finite number of values or a countable number of values. That is, the number of possible values that x can assume is 0, or 1, or 2, etc. ● Acontinuous random variable has infinitely many values and those values are often associated with measurements on a continuous scale with no gaps or interruptions ● In this chapter, continuous probability distributions are presented ● If a continuous random variable has a distribution with a graph that is symmetric and bell-shaped, and can be described by the equation given, we say it has a normal distribution ● The normal distribution is determined by two parameters: the mean, μ, and standard deviation, σ 5.2 The Standard Normal Distribution Uniform Distributions ● Acontinuous variable has a uniform distribution if it’s values spread evenly over the range of possibilities; the graph of a uniform distribution results in a rectangular shape ● The graph of a discrete probability distribution is called a probability histogram; the graph of a continuous probability distribution is called a density curve ○ density curve must satisfy the following properties: ■ total area under the curve must equal 1 ■ every point on the curve must have a vertical height that is 0 or greater ● Due to the total area under the density curve equalling 1, there is a correspondence between the area and probability Standard Normal Distributions ● The density curve of uniform distributions is rectangular, thus easy to find areas/probabilities but for a density curve of normal distribution, it is more complicated due to the bell shape ○ despite the shape, correspondence between area and probability remains ● The standard normal distribution is a normal probability distribution that has a mean of 0 and a standard deviation of 1, and the total area under its density curve is equal to 1 ○ Using the standard normal distribution is easiest for calculations Finding Probabilities When Given z Scores ● z score: the distance along the horizontal scale of the standard normal distribution ● Area: region under the curve ● Notation: ○ P(a < z < b) - denotes probability that z score is between a and b ○ P (z > a) denotes probability that z score is greater than a ○ P (z < a) denotes probability that z score is less than a ● With a continuous probability distribution such as a normal distribution, the probability of getting any single exact value is 0; P(z=a) = 0 ○ e.g. there is a 0 probability of randomly selecting someone and getting a height of 68.12345667 in. ○ In a normal distribution, any single point on a horizontal scale is represented not by a region under the curve, but by a vertical line above the point; a line with no area ● P(a <= z <= b) = P(a < z < b) Finding z Scores from KnownAreas ● Note, z scores positioned in the left half of the curve are always negative ● In cases where we know the area or probability and want to find the corresponding z score, remember to avoid confusion between z scores and areas and following the steps: ○ Draw a bell shaped curve and identify region under the curve that corresponds to given probability ○ Using cumulative area from left, locate the closest probability in the body of TableA-2 and identify the corresponding z score ● Note, when finding the z score and a desired value is midway between two table values, select the larger value to be safe 5.3 Applications of Normal Distributions ● Standard normal distributions with means or 0 and standard deviations of 1 are unrealistic ● However, when working with nonstandard normal distributions, fortunately we can transform its values to a standard normal distribution ● If we convert values to standard scores, then procedures for working with all normal distributions are same as those for the standard normal distribution ○ z= (x - μ) / σ ● The area in any normal distribution bounded by some score x is the same as the area bounded by the equivalent z score in the standard normal distribution Finding Values from KnownAreas ● In many practical and real cases, the area (probability) is known and we must find the relevant value(s); when finding them, be cautious of the following: ○ Don’t confuse z scores and areas ○ Choose the correct side of the graph (right/left) ○ Az score must be negative whenever it is located in the left half of the normal distribution ○ Areas (probabilities) are positive or zero values but they are never negative ● Remember, when rearranging the equation z= (x - μ) / σ into x = μ + (z * σ) to solve for the value x from a known area (and thus known z score), if the z is located to the left of the mean, be sure it is a negative number 5.4 Sampling Distributions and Estimators ● The main objective of this section is to learn what we mean by a sampling distribution of a statistic, and another is to learn a basic principle about sampling distribution of sample means and the samplingt distribution of sample proportions ● It is rare that we know
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