University:
California State University, Northridge
Course:
MATH 340 | Introduction to Probability and Statistics
Academic year:

2009
Views:

243

Pages:

6
Author:

ncshillajjp

Lecture 6: Chapter 6: Normal Probability Distributions A normal distribution is a continuous probability distribution for a random variable x. The graph of a normal distribution is called the normal curve. A normal distribution has the following properties. 1. 2. 3. 4. The mean, median, and mode are equal. The normal curve is bell shaped and is symmetric about the mean. The total are under the normal curve is equal to one. The normal curve approaches, but never touches, the x-axis as it extends farther and farther away from the mean. 5. Between and (in the center of curve) the graph curves downward. The graph curves upward to the left of and to the right of The points at which the curve changes from curving upward to curving downward are called inflection points. 6. The Empirical Rule: Approximately 68% of the area under the normal curve and . Approximately 95% of the area under the is between normal curve is between and area under the normal curve is between . Approximately 99.7% of the and . 6-2 The standard Normal Distribution Def: The standard normal distribution is a normal probability distribution that has a mean of 0 and a standard deviation of 1. Finding Area under the Standard Normal curve using Table A-2: When using table A-2, it is essential to understand the following points. I) II) Table A-2 is designed only for the standard normal distribution, which has a mean of 0 and a standard deviation of 1. Each value in the body of the table is the cumulative area from the left op to a vertical line above a specific value of z. Recall: the z-score allows us to transform a random variable x with mean and standard deviation into a random variable z. with mean 0 and standard deviation of 1. III) IV) When this transformation takes place, the area that falls in the interval under the nonstandard normal curve is the same as that under the standard normal curve with the corresponding z-boundaries. Table A-2 is on two pages. One for positive z scores and one for negative z scores. Example 1: Find the indicated area under the standard normal curve. a) b) c) d) e) f) g) Between z=0 and z = 1.96 To the right of z = 1.64 To the left of z = 1.54 To the right of z = -0.95 To the left of z =-2.57 Between z=-0.44 and z = 1.66 Between z =1.66 and z = 2.97 Notation: P(aa) denotes the probability that the z score is greater than a. Note: P(z>a) = 1 - p(z30, are drawn from any population with a mean and a standard deviation of , then the sampling distribution of sample means approximates a normal distribution. The greater the sample size, the better the approximation. 2. If the population itself is normally distributed, the sampling distribution of sample means is normally distributed for any sample size n. In either case, the sampling distribution of sample means has a mean equal to the population mean. = 3. And the sampling distribution of sample means has a variance equal to 1/n times the variance of the population and a standard deviation equal to the population standard deviation divided by the square root of n. The standard deviation of the sample means is often called the standard error of the mean. Probability and the Central Limit Theorem We can find the probability that a sample mean, , will fall in a given interval of the sampling distribution. To transform to a z score, you can use the equation Example 14: The mean rent of an apartment in a professionally managed apartment building is $780. You randomly select nine professionally managed apartments. What is the probability that the mean rent is less than $825? Assume that the rents are normally distributed, with a standard deviation of $150. Example 15: Credit card balances are normally distributed, with a mean of $2870 and a standard deviation of $900. A) What is the probability that a randomly selected credit card holder has a credit card balance less than $2500? B) You randomly select 25 credit card holders. What is the probability that their mean credit card balance is less than $2500? Example 16: During a certain week the mean price of gasoline in the New England region was 0 per gallon. What is the probability that the mean price for a sample of 32 randomly selected gas stations in that area was between $1.075 and $1.090 that week? Assume the standard deviation is $0.045

Normal Probability Distributions

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