Section 8.5 The Normal Distribution

Transcription

1 Section 8.5 The Normal Distribution Properties of the Normal Distribution Curve 1. We denote the normal random variable with X = x. 2. The curve has a peak at x = µ. 3. The curve is symmetric about the line x = µ. 4. The curve always lies above the x-axis but approaches the x-axis as x extends indefinitely in either direction. 5. The area under the curve is The Standard Normal distribution has µ = 0 and σ = 1. We denote the standard normal random variable with Z = z. Calculating the Probability of a Normal Random Variable The probability P (a < X < b) that X lies between a and b is the area under the curve between x = a and x = b. This can be found using probability tables but in this class we will use the calculator function normalcdf to calculate probabilities for a Normal random variable. Calculator Steps: Click 2ND, VARS, 2. You should see normalcdf( on your screen. The format is normalcdf(smallest x-value/z-value, biggest x-value/z-value, µ, σ). Use E99 if the biggest x-value/z-value is and E99 if the smallest x-value/z-value is. To get E99 click 2ND,,. Note: We never use normalpdf in this class.

2 1. Answer the following: (a) Choose a sketch of the area under the standard normal curve corresponding to P (0.29 < Z < 1). (a) (b) (c) (d) (b) Find the value of the probability of the standard normal variable Z corresponding to P (0.29 < Z < 1). (Give answer to four decimal places.) 2. Find the indicated probability given that Z is a random variable with a standard normal distribution. (Round answer to four decimal places.) P (Z 0.71) 3. Suppose X is a normal random variable with µ = 376 and σ = 17. Find the following probabilities. (Give answers to four decimal places.) (a) P (X < 409) (b) P (389 < X < 411) 2 Fall 2017, Maya Johnson

3 (c) P (X > 409) Inverse Normal Distribution: Suppose we are given the probability or area under the curve and are asked to find the random variable value that corresponds to the given probability. To solve this problem we will use the calculator function invnorm. Calculator Steps: Click 2ND, VARS, 3. You should see invnorm( on your screen. The format is invnorm(probability to the left of X = x or Z = z, µ, σ). 4. Let Z be the standard normal variable. Find the values of a that satisfy the given probabilities. (Give answers to four decimal places.) (a) P (Z > a) = (b) P (Z < a) = Fall 2017, Maya Johnson

4 5. Let Z be the standard normal variable. Find the values of a that satisfy the given probabilities. (Give answers to four decimal places.) (a) P (Z > a) = (b) P ( a < Z < a) = Find the indicated quantities given that X is a normal random variable with a mean of 40 and a standard deviation of 10. (Round answers to four decimal places.) (a) Find the value of b such that P (X b) = (b) Find the value of c such that P (X c) = Fall 2017, Maya Johnson

5 (c) Find the values of A and B such that P (A X B) = if A and B are symmetric about the mean. 5 Fall 2017, Maya Johnson

6 Section 8.6 Applications of the Normal Distribution 1. On the average, a student takes 119 words/minute midway through an advanced court reporting course at the American Institute of Court Reporting. Assuming that the dictation speeds of the students are normally distributed and that the standard deviation is 24 words/minute, find the probability that a student randomly selected from the course can make dictation at the following speeds. (Give answers to four decimal places.) (a) more than 167 words/minute (b) between 143 and 167 words/minute (c) less than 71 words/minute 2. The weight of topsoil sold in a week is normally distributed with a mean of 800 tons and a standard deviation of 32 tons. (Round answers to two decimal places.) (a) What percentage of weeks will sales exceed 864 tons? (b) What percentage of weeks will sales be less than 784 tons? (c) What percentage of weeks will sales be between 752 and 816 tons? 6 Fall 2017, Maya Johnson

7 3. A teacher wishes to curve a test whose grades were normally distributed with a mean of 60 and standard deviation of 15. The top 10% of the class will get an A, the next 30% of the class will get a B, the next 35% of the class will get a C, the next 20% of the class will get a D and the bottom 5% of the class will get an F. Find the cutoff for each of these grades. (Round answers to two decimal places.) (a) The A cutoff is a grade of (b) The B cutoff is a grade of (c) The C cutoff is a grade of (d) The D cutoff is a grade of 7 Fall 2017, Maya Johnson

8 4. TKK Products manufactures 50-, 60-, 75-, and 100-watt electric light bulbs. Laboratory tests show that the lives of these light bulbs are normally distributed with a mean of 700 hr and standard deviation of 75 hr. What is the probability that a TKK light bulb selected at random will burn for the following times? (Round answers to four decimal places.) (a) more than 900 hr (b) less than 600 hr (c) between 700 and 900 hr 5. The distribution of heights of adult males is normally distributed with mean 63 inches and standard deviation 2.4 inches. Answer the following. (Round answers to 2 decimal places.) (a) What minimum height is taller than 64% of all adult males? (b) What two heights that are symmetric about the mean enclose the middle 82% of all heights? 8 Fall 2017, Maya Johnson