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Chapter 6 OutlineIdentify distributions as symmetrical or skewed.Identify the properties of the normal distribution.Find the area under the standard normal distribution, given various z values.Find the probabilities for a normally distributed variable by transforming it into a standard normal variable.Find specific data values for given percentages using the standard normal distribution.Use the central limit theorem to solve problems involving sample means for large samples.Use the normal approximation to compute probabilities for a binomial variable.IntroductionMany continuous variables have distributions that are bell-shaped and are called approximately normally distributed variables.A normal distribution is also known as the bell curve or the Gaussian distribution.Normal and Skewed DistributionsThe normal distribution is a continuous, bell-shaped distribution of a variable.If the data values are evenly distributed about the mean, the distribution is said to be symmetrical.If the majority of the data values fall to the left or right of the mean, the distribution is said to be skewed.Left Skewed DistributionsWhen the majority of the data values fall to the right of the mean, the distribution is said to be negatively or left skewed. The mean is to the left of the median, and the mean and the median are to the left of the mode.Right Skewed DistributionsWhen the majority of the data values fall to the left of the mean, the distribution is said to be positively or right skewed. The mean falls to the right of the median and both the mean and the median fall to the right of the mode.Equation for a Normal DistributionThe mathematical equation for the normal distribution is:Properties of the Normal DistributionThe shape and position of the normal distribution curve depend on two parameters, the mean and the standard deviation.Each normally distributed variable has its own normal distribution curve, which depends on the values of the variable's mean and standard deviation.Normal Distribution PropertiesThe normal distribution curve is bell-shaped.The mean, median, and mode are equal and located at the center of the distribution.The normal distribution curve is unimodal (i.e., it has only one mode).The curve is symmetrical about the mean, which is equivalent to saying that its shape is the same on both sides of a vertical line passing through the center.Normal Distribution Properties (cont'd.)The curve is continuous--i.e., there are no gaps or holes. For each value of X, here is a corresponding value of Y.The curve never touches the x axis. Theoretically, no matter how far in either direction the curve extends, it never meets the x axis--but it gets increasingly closer.Normal Distribution Properties (cont'd.)The total area under the normal distribution curve is equal to 1.00 or 100%.The area under the normal curve that lies within one standard deviation of the mean is approximately 0.68, or 68%; within two standard deviations, about 0.95, or 95%; and...

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