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7.4 Use Normal Distributions 7.3-7.4 HW Quiz: August 18 7.3-7.4 Quiz: August 20 Vocabulary • Normal Distribution Is modeled by a bell-shaped curve called a normal curve that is symmetric about the mean. • The total area under the related curve is 1. • The percentage of the area covered by each standard deviation from the mean is shown in the graph. (on the next slide). Graph of a Normal Curve: σ=1 standard deviation Vocabulary • Standard Normal Distribution Is the normal distribution with mean 0 and standard deviation 1. Example 1: • A normal distribution has mean x and standard deviation σ. Find the indicated probability for a randomly selected x-value from the distribution. P(x > x + σ) P(x < x < x + σ) You Try: • A normal distribution has mean x and standard deviation σ. Find the indicated probability for a randomly selected x-value from the distribution. P(x < x + σ) P(x < x < x + 2σ) Example 2: • The heights (in feet) of fully grown white oak trees are normally distributed with a mean of 90 feet and a standard deviation of 3.5 feet. – About what percent of white oak trees have heights between 86.5 feet and 93.5 feet? Formula • The formula below can be used to transform x-values from a normal distribution with mean x and standard deviation σ into z-values having a standard normal distribution. Z-Value • The z-value for a particular x-value is called the z-score for the x-value and is the number of standard deviation the x-value lies above or below the mean x. ***To use the z-score you will need to look at the table that is on p. 296 in your textbook*** Example 3: • Use the information about white oak trees from Example 2 to answer the following. – Find the probability that a randomly selected white oak tree has a height of at most 94 feet. – You need to find the z-score (use the formula) – Now you need to use the Standard Normal Table You Try: • In the following exercises, refer to Example 2. 1. About what percent of white oak trees have height between 83 feet and 90 feet. 2. Find the probability that a randomly selected white oak tree has a height of at most 85 feet. Homework P. 266 #1-19odd