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Lecture 13 Standard Normal Distribution Continuous Probability Distribution This chapter continues our study of probability distributions by examining the continuous probability distribution. Recall that a continuous probability distribution can assume an infinite number of values within a given range. As an example, the weights for a sample of small engine blocks are: 54.3, 52.7, 53.1 and 53.9 pounds. The normal probability distribution is described by its mean and standard deviation. Suppose the life of an automobile battery follows the normal distribution with a mean of 36 months and a standard deviation of 3 months. We can determine the probability that a battery will last between 36 and forty months. Life of a battery is measured on a continuous scale. Normal Probability Distribution A continuous probability distribution uniquely determined by µ and σ. The Greek letter µ (mu), represents the mean of a normal distribution and the Greek letter σ (sigma) represents the standard deviation. The major characteristics of the normal distribution are: 1. The normal distribution is "bell-shaped" and the mean, median, and mode are all equal and are located in the center of the distribution. Exactly one-half of the area under the normal curve is above the center and one-half of the area is below the center. 2. The distribution is symmetrical about the mean. A vertical line drawn at the mean divides the distribution into two equal halves and these halves possess exactly the same shape. 3. It is asymptotic. That is, the tails of the curve approach the X-axis but never actually touch it. 4. A normal distribution is completely described by its mean and standard deviation. This indicates that if the mean and standard deviation are known, a normal distribution can be constructed and its curve drawn. 5. There is a "family" of normal probability distributions. This means there is a different normal distribution for each combination of µ and σ. Instructor: Ms. Azmat Nafees 15-Oct-09 These characteristics are summarized in the graph. The Standard Normal Probability Distribution Since there are an infinite number of probability distributions, it would be awkward to construct tables of probabilities for so many different normal distributions. An efficient method for overcoming this difficulty is to standardize each normal distribution. Therefore, a normal distribution with a mean of 0 and a standard deviation of 1 is known as standard normal distribution. An actual distribution is converted to a standard normal distribution using a z value. The formula for a specific standardized z value is text formula: Where: X is the value of any particular observation or measurement. µ is the mean of the distribution. σ is the standard deviation of the distribution. z is the standardized normal value, usually called the z value. Applications of the Standard Normal Distribution To obtain the probability of a value falling in the interval between the variable of interest (X) and the mean (µ), we first compute the distance between the value (X) and the mean (µ). Then we express that difference in units of the standard deviation by dividing (X - µ) by the standard deviation. This process is called standardizing. Example 1: Suppose the mean useful life of a car battery is 36 months, with a standard deviation of 3 months. What is the probability that such a battery will last between 36 and 40 months? z 0.00 0.01 0.02 0.03 0.04 0.05 ! ! ! ! ! ! ! ! 1.0 1.1 The first step is to convert the 40 months to an equivalent standard normal value, using formula [7–5]. The computation 1.2 0.3665 0.3686 0.3708 0.3729 1.3 0.4049 0.4066 0.4082 0.4099 1.4 0.4207 0.4222 0.4236 0.4251 is: Instructor: Ms. Azmat Nafees 0.3869 0.3888 0.3907 0.3925 15-Oct-09 Next refer to a table for the areas under the normal curve. A part of the table is shown at the right. To use the table, the z value of 1.33 is split into two parts, 1.3 and 0.03. To obtain the probability go down the left-hand column to 1.3, then move over to the column headed 0.03 and read the probability. It is 0.4082. The probability that a battery will last between 36 and 40 months is 0.4082. Example 2: The mean weekly income of a shift foreman in the glass industry is normally distributed with a mean of $1,000 and a standard deviation of $100. (a) What is the likelihood of selecting a foreman whose weekly income is between $1,000 and $1,100? (b) What is the probability of selecting a shift foreman in the glass industry whose income is between $790 and $1,000? (c) What is the probability of selecting a shift foreman in the glass industry whose income is between $840 and $1,200? Empirical Rule Before examining various applications of the standard normal probability distribution, three areas under the normal curve will be considered. 1. About 68 percent of the area under the normal curve is within plus one and minus one standard deviation of the mean. This can be written as µ ± 1σ. 2. About 95 percent of the area under the normal curve is within plus and minus two standard deviations of the mean, written µ ± 2σ. 3. Practically all of the area under the normal curve is within three standard deviations of the mean, written µ ± 3σ. The estimates given above are the same as those shown on the diagram. Example 3: The daily water usage per person in Gulberg, Lahore is normally distributed with a mean of 80 litres and a standard deviation of 10 litres. About 68 percent of those living in Gulberg will use how many litres of water? About 68% of the daily water usage will lie between 70 and 90 litres (using µ ± 1σ). Instructor: Ms. Azmat Nafees 15-Oct-09