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Chapter 7 Normal Curves and Sampling Distributions Understanding Basic Statistics Fifth Edition By Brase and Brase Prepared by Jon Booze The Normal Distribution • A continuous distribution used for modeling many natural phenomena. • Sometimes called the Gaussian Distribution, after Carl Gauss. • The defining features of a Normal Distribution are the mean, µ, and the standard deviation, σ. Copyright © Cengage Learning. All rights reserved. 7|2 The Normal Curve Copyright © Cengage Learning. All rights reserved. 7|3 Features of the Normal Curve • Smooth line and symmetric around µ. • Highest point directly above µ. • The curve never touches the horizontal axis in either direction. • As σ increases, the curve spreads out. • As σ decreases, the curve becomes more peaked around µ. • Inflection points at µ ± σ. Copyright © Cengage Learning. All rights reserved. 7|4 Two Normal Curves Both curves have the same mean, µ = 6. Curve A has a standard deviation of σ = 1. Curve B has a standard deviation of σ = 3. Copyright © Cengage Learning. All rights reserved. 7|5 Normal Probability • The area under any normal curve will always be 1. • The portion of the area under the curve within a given interval represents the probability that a measurement will lie in that interval. Copyright © Cengage Learning. All rights reserved. 7|6 The Empirical Rule Copyright © Cengage Learning. All rights reserved. 7|7 The Empirical Rule Copyright © Cengage Learning. All rights reserved. 7|8 The Empirical Rule The masses of the adult ostriches at the Vilas Zoo are normally distributed with a mean of 124 kg and a standard deviation of 10 kg. What is the probability that a randomly-selected ostrich will have a mass between 124 kg and 154 kg? a). 0.15% b). 99.85% c). 49.85% d). 47.5% Copyright © Cengage Learning. All rights reserved. 7|9 The Empirical Rule The masses of the adult ostriches at the Vilas Zoo are normally distributed with a mean of 124 kg and a standard deviation of 10 kg. What is the probability that a randomly-selected ostrich will have a mass between 124 kg and 154 kg? a). 0.15% b). 99.85% c). 49.85% d). 47.5% Copyright © Cengage Learning. All rights reserved. 7 | 10 Raw Scores and z Scores Copyright © Cengage Learning. All rights reserved. 7 | 11 Raw Scores and z Scores For a distribution with = 10 and = 2.5, find the z score of the value x = 15. a). z = 2 b). z = 12.5 c). z = 4 d). z = 3.16 Copyright © Cengage Learning. All rights reserved. 7 | 12 Raw Scores and z Scores For a distribution with = 10 and = 2.5, find the z score of the value x = 15. a). z = 2 b). z = 12.5 c). z = 4 d). z = 3.16 Copyright © Cengage Learning. All rights reserved. 7 | 13 Distribution of z-Scores • If the original x values are normally distributed, so are the z scores of these x values. – µ=0 – σ=1 Copyright © Cengage Learning. All rights reserved. 7 | 14 Using the Standard Normal Distribution There are extensive tables for the Standard Normal Distribution. • We can determine probabilities for normal distributions: 1) Transform the measurement to a z score. 2) Utilize Table 3 of the Appendix. Copyright © Cengage Learning. All rights reserved. 7 | 15 Using the Standard Normal Table • Table 3(a) gives the cumulative area for a given z value. • When calculating a z Score, round to 2 decimal places. • For a z Score less than –3.49, use 0.000 to approximate the area. • For a z Score greater than 3.49, use 1.000 to approximate the area. Copyright © Cengage Learning. All rights reserved. 7 | 16 Area to the Left of a Given z Value Copyright © Cengage Learning. All rights reserved. 7 | 17 Area to the Right of a Given z Value Copyright © Cengage Learning. All rights reserved. 7 | 18 Area Between Two z Values Copyright © Cengage Learning. All rights reserved. 7 | 19 Using a z Table Using Table 3 in the Appendix , find the probability that z > 0.9. a). 0.22 b). 0.09 c). 0.65 d). 0.18 Copyright © Cengage Learning. All rights reserved. 7 | 20 Using a z Table Using Table 3 in the Appendix , find the probability that z > 0.9. a). 0.22 b). 0.09 c). 0.65 d). 0.18 Copyright © Cengage Learning. All rights reserved. 7 | 21 Normal Probability Final Remarks • The probability that z equals a certain number is always 0. – P(z = a) = 0 • Therefore, < and ≤ can be used interchangeably. Similarly, > and ≥ can be used interchangeably. – P(z < b) = P(z ≤ b) – P(z > c) = P(z ≥ c) Copyright © Cengage Learning. All rights reserved. 7 | 22 Inverse Normal Distribution • Sometimes we need to find an x or z that corresponds to a given area under the normal curve. – In Table 3, we look up an area and find the corresponding z. Copyright © Cengage Learning. All rights reserved. 7 | 23 Finding z Corresponding to a Given Area A (0 < A < 1) Left-tail case: the given area is to the left of z. Look up the number A in the body of the table and use the corresponding z value. Copyright © Cengage Learning. All rights reserved. 7 | 24 Finding z Corresponding to a Given Area A (0 < A < 1) Right-tail case: the given area is to the right of z. Look up the number 1 – A in the body of the table and use the corresponding z value. Copyright © Cengage Learning. All rights reserved. 7 | 25 Finding z Corresponding to a Given Area A (0 < A < 1) Center-tail case: the given area is symmetric and centered above z = 0. Look up the number (1 – A)/2 in the body of the table and use the corresponding ±z value. Copyright © Cengage Learning. All rights reserved. 7 | 26 Inverse Normal Distribution Using Table 3 in the Appendix, find the range of z scores, centered about the mean, that contain 70% of the probability. a). –1.04 to 1.04 b). –2.17 to 2.17 c). –0.30 to 0.30 d). –0.52 to 0.52 Copyright © Cengage Learning. All rights reserved. 7 | 27 Inverse Normal Distribution Using Table 3 in the Appendix, find the range of z scores, centered about the mean, that contain 70% of the probability. a). –1.04 to 1.04 b). –2.17 to 2.17 c). –0.30 to 0.30 d). –0.52 to 0.52 Copyright © Cengage Learning. All rights reserved. 7 | 28 Critical Thinking – How to tell if data follow a normal distribution? • Histogram – a normal distribution’s histogram should be roughly bell-shaped. • Outliers – a normal distribution should have no more than one outlier Copyright © Cengage Learning. All rights reserved. 7 | 29 Critical Thinking – How to tell if data follow a normal distribution? • Skewness –normal distributions are symmetric. Use the Pearson’s index: 3( x median) Pearson’s index = s A Pearson’s index greater than 1 or less than –1 indicates skewness. • Normal quantile plot – using a statistical software (see the Using Technology feature.) Copyright © Cengage Learning. All rights reserved. 7 | 30 Terms, Statistics & Parameters • Terms: Population, Sample, Parameter, Statistics Copyright © Cengage Learning. All rights reserved. 7 | 31 Why Sample? • If time and resources are limited, we take samples to learn about the population. Copyright © Cengage Learning. All rights reserved. 7 | 32 Types of Inference 1) Estimation: We estimate the value of a population parameter. 2) Testing: We formulate a decision about a population parameter. 3) Regression: We make predictions about the value of a statistical variable. Copyright © Cengage Learning. All rights reserved. 7 | 33 Sampling Distributions • To evaluate the reliability of our inference, we need to know about the probability distribution of the statistic we are using. • Typically, we are interested in the sampling distributions for sample means and sample proportions. Copyright © Cengage Learning. All rights reserved. 7 | 34 The Central Limit Theorem (Normal) • If x is a random variable with a normal distribution, mean = µ, and standard deviation = σ, then the following holds for any sample size: (n is the sample size) Copyright © Cengage Learning. All rights reserved. 7 | 35 The Standard Error • The standard error is just another name for the standard deviation of the sampling distribution. Copyright © Cengage Learning. All rights reserved. 7 | 36 The Central Limit Theorem (Any Distribution) • If a random variable has any distribution with mean = µ and standard deviation = σ, the sampling distribution of x will approach a normal distribution with mean = µ and standard deviation = n as n increases without limit. Copyright © Cengage Learning. All rights reserved. 7 | 37 Sample Size Considerations • For the Central Limit Theorem (CLT) to be applicable: – If the x distribution is symmetric or reasonably symmetric, n ≥ 30 should suffice. – If the x distribution is highly skewed or unusual, even larger sample sizes will be required. – If possible, make a graph to visualize how the sampling distribution is behaving. Copyright © Cengage Learning. All rights reserved. 7 | 38 Critical Thinking • Bias – A sample statistic is unbiased if the mean of its sampling distribution equals the value of the parameter being estimated. • Variability – The spread of the sampling distribution indicates the variability of the statistic. Copyright © Cengage Learning. All rights reserved. 7 | 39 Normal Approximation to the Binomial Copyright © Cengage Learning. All rights reserved. 7 | 40 Normal Approximation to the Binomial A fair coin is flipped 200 times and the number of heads, x, is counted. Find the normal approximation of the standard deviation for this experiment. a). 50 b). 7.07 Copyright © Cengage Learning. All rights reserved. c). 10 d). 100 7 | 41 Normal Approximation to the Binomial A fair coin is flipped 200 times and the number of heads, x, is counted. Find the normal approximation of the standard deviation for this experiment. a). 50 b). 7.07 Copyright © Cengage Learning. All rights reserved. c). 10 d). 100 7 | 42 Continuity Correction Copyright © Cengage Learning. All rights reserved. 7 | 43