The normal distribution is a theoretical model derived mathematically and not empirically.
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1 Sociology 541 The Normal Distribution Probability and An Introduction to Inferential Statistics Normal Approximation The normal distribution is a theoretical model derived mathematically and not empirically. The following equation describes the normal distribution: 1 f ( x) = e σ 2π 2 2 ( x µ ) /(2σ ) Characteristics of normal distribution a. bell-shaped b. symmetrical c. mean, mode and median coincide e. asymptotic distribution f. total area under curve equals 1 This is an extremely useful distribution because some variables in empirical world have frequency distributions that are similar in form to the normal curve. Note from the equation describing a normal distribution that normal distributions depend only on the mean µ and the standard deviation σ. There are an infinite number of normal distributions, one for each combination of possible means and standard deviations. Example of two normal distributions These distributions are both normal but share a different mean and standard deviation. However, we can convert these distributions into standard scores or Z scores and express them in the same scale. When a normal curve has been converted into Z scores with a mean of zero and standard deviation of one, it is called the standard normal distribution. 1
2 Normal curve Mean is zero, standard deviation is 1 All normal distributions share an important characteristic: A given standard deviation from the mean always cuts off the same proportion or percentage of scores in all normal distributions. The area under the normal curve x +/- 1s: 68% of all observations are within 1 standard deviation unit below and 1 s above mean x +/- 2s: 95% of all observations are within 2 standard deviation unit below and 2 s above mean x +/- 3s: 99% of all observations are within 3 standard deviation unit below and 3 s above mean Theoretical model of bell-shaped curve is useful because it is known that 68% of all observations are within 1 standard deviation below and above mean, 95% are within two standard deviations below and above mean, and 99% below and above mean. Reading a Z table You have a set of observations that come from a normal distribution and want to find out the percentage of those observations that are more than 1.5 standard deviations above the mean: Z=1.5 What if you want to find the percentage of scores that fall within 0.75 standard deviations of the mean? Z=0.75 Percentage of scores that are more than 1.3 standard deviations away from the mean? Z=1.3 Example: Time to complete college achievement test was approximately normal with an average of 110 minutes and standard deviation of 20 minutes. 2
3 What proportion finished in 2 hours or less? What time is in the 80 th percentile? Probability and An Introduction to Inferential Statistics Probability comes into play when we talk about inferential statistics and how to generalize from sample findings to populations. Statistical inference makes it possible to estimate or infer a population parameter from a sample statistic. Probability is used to assess the likelihood that an observed sample statistic represents a population parameter. We ll distinguish between three types of distributions: A sample distribution A population (probability) distribution A sampling distribution PROBABILITY Definition: The probability of a particular outcome is the proportion of times that outcome would occur in a long run of repeated observations. Another way to think of the probability associated with an event is that it is the expected number of times that an event will occur relative to the total number of times any event can occur. Notation: P(x) = Means that the probability of the outcome x occurring is Examples Sample Distribution The distribution of data (drawn from a sample) that we actually observe. Probability Distribution A probability distribution is directly analogous to a frequency distribution, except that it is based on theory (probability theory) rather than on what is observed in the real world (empirical data). 3
4 In a probability distribution, we specify the possible values of a variable and calculate the probabilities associated with each. Probability Distributions for Discrete Outcomes Die Toss Outcome Probability Coin Toss Notation Let x denote a possible outcome for variable X and let P(x) denote the probability of that outcome. 0<= P(x) <= 1 Σ P(x) = 1. We can calculate a mean (expected value) for a probability distribution. µ = Σ x P(x) Mean for Die Roll: Mean for Coin Toss: Plotting probability distributions Just as we have done with relative frequency distributions, we can use bar graphs to plot the probability distribution of a discrete variable. Rectangular bar over a possible value of the variable has height equal to the probability of that value. These probability distributions (or population distributions) can be distinguished from sample distributions. In reality, we don't observe the probability or population distribution. We only observe a sample distribution. 4
5 The probability distribution represents the expected values or outcomes in an infinite number of trials. What would a probability distribution look like if you had only a limited number of trials? If it were based on a sample of fixed size n? To find the chance, we must find all the possible ways this event could happen in a sample of size n, calculate the chance or probability of each and add up all these probabilities/chances. Example of coin toss N=1 X = Number of Heads Two possible outcomes: H or T chance of each X Prop(X) P(X) N=2 X = Head Four possible outcomes: HH, HT, TH, TT X Prop(X) F P(X) N = Number of coin tosses X = Number of heads (outcome of interest) F = Number of ways that you can X when N=2 P(X) = Probability (relative frequency) of each outcome Prop (X) = Proportion of heads (X / N) N=3 X = Head N=4 X = Head Sixteen possible outcomes: (2) 4 5
6 We refer to these distributions as sampling distributions. A sampling distribution is a probability distribution that determines probabilities of the possible values of a sample statistic. We can display these four sampling distributions (for samples of sizes n=1 to 4) graphically. Graph below shows sampling distribution of sample proportion of heads, for sample of n=4 from population with true proportion of Mean PROB H 1H 2H 3H.1 4H OUTCOME What do you notice about the shape of these sampling distributions as n increases? 6
7 Sampling Distribution of Sample Means The sampling distribution of sample means has three important characteristics. 1. The distribution of sample means will approximate the form of a normal distribution if the size of each of the samples is equal to or greater than 30. (Central Limit Theorem) 2. The mean of the distribution of sample means (µ x ) equals the real population mean (µ). If the population mean is the mean of the sampling distribution and the sampling distribution approximates the normal distribution, we can start to make statements about how likely it is to observe a particular sample mean. Any sample mean you encounter, regardless of which original measure or distribution produced the mean, comes from a normal distribution. 3. The standard deviation (standard error) of the sampling distribution (σ x ) has fixed relationship to the standard deviation of the population (σ). This enables us to determine the amount of error due to sampling variability. 7
8 Using the Binomial Formula When a variable has a dichotomous (2) outcome, the Bernoulli frequency function can be used to determine the probabilities for particular outcomes. The chance that an event will occur exactly x times out of n is given by the binomial formula (Bernoulli Frequency Function): P( X x n p x p x n x = ) = (1 ) n! x!( n x)! x n x = p (1 p) In this formula, n is the number of trials, x is the number of times the event is to occur, and p is the probability that the event will occur on any particular trial. We assume: The value of n must be fixed in advance. P must be the same from trial to trial. The trials must be independent. Using the Binomial formula, what s the probability of only one head in four coin tosses? What s the probability of no heads? Now, assume that we toss a coin ten times. What is the probability of obtaining just one head in 10 tosses? Can use Bernoulli's frequency function or can use Table B.2 to look up the probabilities associated with various sample sizes and outcomes. Appendix B: Table B.2: Areas under the Binomial Curve Using the table, construct a probability distribution for proportion of heads in ten tosses. 8
9 Properties of the Binomial Distribution: 1. The probabilities sum to When p=q=1/2, the probability distribution is symmetrical. 3. As n increases, the shape of the distribution of outcomes begins to approximate that of the normal distribution. (Approaches normal distribution faster when p=q=1/2). SUMMARY 1. Population distribution: distribution from which we select the sample. Usually unknown and we make inferences about its characteristics, µ and σ. 2. Sample distribution: distribution of data that we actually observe, x and s. The larger the sample size, the closer the sample distribution resembles the population distribution. 3. Sampling distribution: probability distribution of a sample statistic, such as x. Describes variability that occurs in the value of the statistic among samples of a certain size drawn from the same population (denoted by µ x and σ x ). Determines the probability that the statistic falls within a certain distance of the population parameter. 9
10 COIN TOSS EXERCISE In this example, the probability distribution for a coin toss is equivalent to a population distribution. The experiments of 10 coin tosses is equivalent to samples of size 10 drawn from the population. For a single toss of a coin, let Y=1 for a head and Y=0 for a tail. a. Assuming the coin is balanced, construct the probability distribution for Y and calculate its mean. b. The coin is flipped ten times, yielding six heads and four tails. Construct a sample relative frequency distribution. c. Run four separate samples (experiments) of 10 tosses (trials) each and construct a sample relative frequency distribution for each of the four samples. What is the proportion of heads in each sample? d. Summarize the empirical sampling distribution by constructing a histogram of all the proportion of heads obtained from each sample. Y F e. If we performed the experiment in c and d an indefinitely large number of times, what values would we get for the mean of the sample proportion values? Recommended Assignment: (To be done in Excel or SPSS) Take a population with a binomial distribution. Assume that the probability of outcome X (P(X)) equals 0.9. Determine the probability distribution of samples with sizes N=10, N=30, and N=100 drawn from this population using the Bernoulli frequency function. Plot a bar graph of these three probability distribution of X (P(X)). What happens to the shape of the distribution as n increases? 10
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