Probability Distributions
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1 Unit 5 Probability Distributions Section 7.3A: Applications of the Normal Distribution Notes By the end of this lesson, you will be able to Find and interpret the area under a normal curve Find the value of a normal random variable. Name: Date: Hour: We want to calculate probabilities and values for normal probability distributions. We know that the area under the curve represents the probability. By calculating the z-score for a specific random variable, X, we can determine the area under the standard normal curve. By finding the area under the standard normal curve, we are finding the area under any normal curve. The following method can be used to compute areas for any normal random variable X Reminder: Step 1: Draw a normal curve and shade the desired area. Step 2: Convert all values of X to Z-scores using Z = X μ σ. Step 3: Find the area under the standard normal curve using your calculator (normaldcf) < means less than > means more than Note: This is the same thing we did last week except we first need to convert the data values into z-scores!
2 Example 1: Assume the random variable X is normally distributed with mean = 50 and standard deviation of 7. a. Compute P(X < 35) b. Compute P(40 < X < 65) Example 2: Assume that a set of data is normally distributed with μ = 14 and σ = 1.5. a. Find the RANDOM VARIABLE (not z-score) that has 14% of the data to it s left. b. Find the RANDOM VARIABLE (not z-score) that has 13% of the data to it s right.
3 Quick Check Section 7.3A: Applications of the Normal Distribution Self-Assessment 1. Assume a random variable has a mean = 22 and a standard deviation of 4.3 a. Compute P(X>27). b. Find the value of the RANDOM VARIABLE that has 56% of the data to it s right. Learning Goals Self-Assessment I am unsure of or confused about this I am ready to start practicing I am already good at this Find and interpret the area under a normal curve Find the value of a normal random variable. My Goals for Today- thinking about what I am good at, where am I confused and what do I need to work on? What do I do if I am confused or need to work on a learning target?
4 Name: Date: Hour: Unit 5 Probability Distributions Section 7.3A: Applications of the Normal Distribution Homework 1. Assume the random variable X is normally distributed with mean = 50 and standard deviation=7. Compute the following probabilities. BE SURE TO DRAW A NORMAL CURVE WITH THE AREA CORRESPONDING TO THE PROBABILTY SHADED. a. P(X>35) b. P(X<45) c. P(40<X<65) d. P(X<55 or X>60) 2. The mean incubation time of fertilized chicken eggs kept at 100.5ºF in a still-air incubator is 21 days. Suppose the incubation times are approximately normally distributed with a standard deviation of 1 day. a. What is the probability that a randomly selected fertilized chicken egg hatches in less than 20 days? b. What is the length of time (day) that has 43% of data to it s left?
5 3. A set of data is approximately randomly distributed with a mean of and a standard deviation of a. Find the random variable that corresponds to the 20 th percentile (that is, find the random variable that separates the bottom 20% from the top 80%). b. Find the random variable that corresponds to the 33 rd percentile. c. Find the random variable that corresponds to the 98 th percentile. d. Find the random variable that has 46% of the data to it s right. e. Find the random variable that has 77% of the data to it s right.
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