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This 4 page Class Notes was uploaded by Jillian Holmes on Monday October 3, 2016. The Class Notes belongs to Stat 206 at University of South Carolina taught by Professor Ward-Besser in Fall 2016. Since its upload, it has received 17 views.
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Date Created: 10/03/16
STAT 206 Chapter 6: Normal Distribution and Other Continuous Distributions 6.2 The Normal Distribution ● Normal Distribution: ost common continuous distribution used in statistics ○ Importance: ■ Numerous continuous variables common in business have distributions that closely resemble the normal distribution ■ The normal distribution can be used to approximate various discrete probability distributions ■ The normal distribution provides the basis lassical statistical inference because of its relationship t entral Limit Theorem ○ Theoretical Properties: ■ Bell-shaped curve ■ Symmetrical ■ Mean, median, and mode are equal ■ Its interquartile range is equal to 1.33 standard deviations. Thus, the middle 50% of the values are contained within an interval of two-thirds of a standard deviation below the mean and two-thirds of a standard deviation above the mean. ■ It has an infinite range (-∞ < X < ∞) ● Probability Density Function for Normal Distribution: ○ Where: ■ e = the mathematical constant approximated by 2.71828 ■ ???? = mathematical constant approximated by 3.14159 ■ μ = the mean ■ ???? = the standard deviation ■ X = any value of the continuous variable, where -∞ < X < ∞ ● Standardized Normal Variable = Z ○ Any normally distributed variable (X) can be transformed i tandardized normal variable (Z) using the transformation formula ○ Transformation Formula: **Z score = standard score** ○ For the standardized normal distribut ean = 0 & standard deviation = 1 ● The standard normal curve is a normal curve with: ○ Mean (μ) = 0 ○ Variance (σ²) = 1 ○ Standard deviation (σ) = 1 ○ Denoted by N(μ, σ²) = N(0,1) ○ hen you calculate z-score, you are transforming your normal curve into a standard normal curve ○ Values above the mean have positive Z-values ○ Values below the mean have negative Z-values ● Excel function for Z-value: STANDARDIZE(X, μ, σ) ● Excel function for normal probability: NORM.DIST(X, μ, σ, TRUE) ● General Procedures for Finding Normal Probabilities ○ To find P(a < X <b) when X is distributed normally: ■ Draw the normal curve for the problem in terms of X ■ Translate X-values to Z-values ■ Use the standardized normal table ● When given a Normal Probability and you need to find x ○ Use the Z = (X-μ) / σ ○ Solve for X ■ Multiply each side by σ → Z(σ) = X-μ ■ Add μ to each side →Zσ+μ = X ○ X = Zσ+μ 6.3 Evaluating Normality ● How to determine the normality: ○ Construct charts and observe their appearance. For small- or moderate-sized data sets, create a stem-and-leaf display or a boxplot. For large data sets, in addition, plot a histogram or polygon. Look for a bell-shaped curve. ○ Compute descriptive statistics and compare these statistics with the theoretical properties of the normal distribution. Compare the mean and median. Is the interquartile range approximately 1.33 times the standard deviation? Is the range approximately 6 times the standard deviation? ○ Evaluate how the values are distributed. Determine whether approximately two-thirds of the values lie between the mean and ± 1 standard deviation. Determine whether approximately four-fifths of the values lie between the mean and ± 1.28 standard deviations. Determine whether approximately 19 out of every 20 values lie between the mean and ± 2 standard deviations. ● Normal Probability Plot: a visual display that helps you evaluate whether the data are normally distributed ○ Constructing a normal probability plot ■ Arrange the data into ordered array ■ Find the corresponding standardized normal quantile values (Z) ■ Plot the pairs of points with observed data values (X) on the vertical axis and the standardized normal quantile values (Z) on the horizontal axis ■ Evaluate the plot for evidence of linearity ○ Evaluating the normal probability plot ■ If the plot is linear, the data is from a normal distribution ■ If the plot rises rapidly at first then levels off, the data is left-skewed ■ If the plot rises slowly at first then more rapidly, the data is right-skewed Right-skewed Left-skewed 6.4 The Uniform Distribution ● Uniform Distribution: a probability distribution that has equal probabilities for all possible outcomes of the random variable; in it, the values are evenly distributed in the range between the smallest value, , and the largest valu ; sometimes called the rectangular distribution ○ Where: ■ f(x) = value of the density function at any X value ■ a = minimum value of X ■ b = maximum value of X 6.6 The Normal Approximation to the Binomial Distribution ● The binomial distribution is a discrete distribution, but the normal is continuous ● To use the normal to approximate the binomial, accuracy is improved if you use a correction for continuity adjustment ● The closer π is to 0.5, the better the normal approximation to the binomial ● The larger the sample size n, the better the normal approximation to the binomial ● General Rule ○ The normal distribution can be used to approximate the binomial distribution if: ■ nπ ≥ 5 and n(1-π) ≥ 5 ○ Remember: mean and standard deviation for binomial ■ μ = nπ ■ σ = √nπ(1-π) ○ If using normal approximation, calculate Z score using the formula: ■ z = (x-μ) / σ = (x-nπ) / √nπ(1-π)
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