Chapter 6 -- The Normal Distribution

Chapter 5 -- The Normal Distribution In the world around us, we observe a wide variety of variables – height, weight, test scores, life of a brand of batteries, etc. Surprisingly, these variables share an important characteristic: their distributions have roughly the shape of a bell-shaped curve (normal curve) like the one shown below. A continuous random variable X is said to be normally distributed or to have a normal distribution if its distribution has the shape of a normal curve. A normal distribution is completely determined by its mean and standard deviation. The probability density function of a normal random variable X is of the form f(x) =

1



e 2

( x   )2

/ 2 2

Where  = mean = E(X) and  = S.D(X) Notation: X ~ N(, ) How to find probabilities using the graph of f?

Standard Normal Random Variable A normal random variable with a mean of 0 and a standard deviation of 1. Such a variable is denoted by Z. That is, Z ~ N(0, 1). Using the TI-83 to find probabilities Example 1: Let Z ~ N(0, 1). Find the following probabilities: 1. 2. 3. 4. 5. 6.

P(Z  -1.15) P(Z  1.15) P(0  Z  0.83) P(-2.45  Z  1.36) P(Z  -0.34) P(Z < -1.5 or Z > 1.5)

0.1251 0.1251 0.2967 0.9060 0.6331 0.1336

Example 2: Let Z ~ N(0, 1). Find z if 1. P(Z < z) = 0.0594 2. P(Z > z) = 0.5 3. P(Z > z) = 0.1635

-1.56 0.0 0.98

Example 3: Let X ~ N(240, 30). Find the following probabilities: 1. P(X  200) 2. P(X  300) 3. P(200  X  300) Example 4: The personnel manager of a large company requires job applicants to take a certain test and achieve a score of at least 500. If the test scores are normally distributed with a mean of 485 and standard deviation of 30, what percentage of the applicants pass the test? (0.3085) Example 5: Experience indicates that the development time for a photographic printing paper is normally distributed with a mean of 30 seconds and standard deviation of 1.1 seconds. Find the probability that it will take between 28.5 and 31.2 seconds for a randomly selected piece of photographic printing paper to develop. (0.7760)

Example 6: The diameter of ball bearings manufactured at a factory is normally distributed with a mean of 3 mm and a standard deviation of 0.1 mm. A customer has specification that require that ball bearings have diameter between 2.85 and 3.1 mm. (a) What fraction of ball bearings manufactured meet specifications? (b) What fraction of ball bearings manufactured do not meet specifications? (0.7745, 0.2255) Grading by the Curve (will discuss this in class) HW: (5.2-5.4) 9-31 (odd) pp. 219-220; 7-15(odd) pp. 225-226; 1, 5, 9, 13, 17, 21, 25, 27, 29, 31, and 33 pp. 234

5.5 The Central Limit Theorem

Population – the set of ALL elements of interest for a particular study. Sample – A subset of a population, usually chosen so that it is representative of population. Reasons for Sampling If you want information about a population, two alternatives are available: 1. Census – sample entire population Exact, takes time, and costly 2. Sample Inexact, quicker, and less expensive. Sampling Error: error resulting from using a sample to estimate a population characteristics. Sampling Distribution of the Mean: For a variable X, and a given sample size n, the distribution of the variable X (that is, of all possible sample means) is called the sampling distribution of the mean. Notations: N= Population size, n = Sample size,

X = = =  =  = X

X

RV associated with the population, Population Mean, Population S. D, Mean of X = E( X ), S.D. of X .

THEOREM 1.  = 2.  = X

X

 /

n

Note: The standard deviation of X (/ n ) is also called the standard error of the mean.

Shape of the Distribution of X . 1. Suppose that a variable X of a population is normally distributed with a mean of  and a standard deviation of . Then, for samples

of size n, the variable X is also normally distributed with a mean of  and a standard deviation of / n . That is, X ~ N(, / n ) 2. (The Central Limit Theorem) For a relatively large sample size, the variable X is approximately normally distributed, regardless of the distribution of X. The approximation becomes better and better with increasing sample size. Note: A sample size of 30 or more is generally considered large.

Example 7: Weights of men are normally distributed with a mean of 170 lbs and a standard deviation of 20 lbs. A sample of 100 men is taken and their weights were observed (a) What is the distribution of X , the sample average? (b) Find the probability that a

randomly selected man will weigh less than 166 lbs? (c) What is the probability that the average weight for the sample selected will be less than 166 lbs? (d) What is the standard error of the mean? Ans. (a) N(170, 2) (b) 0.4207 (c) 0.0228 (d) 2 Example 8: An electrical component is designed to provide a mean service life of 3000 hours with a standard deviation of 800 hours. A customer purchases a batch of 50 components. What is the probability that the mean life for this sample will be a least 2750 hours? Ans. (0.9864)

Example 9: One hundred small bolts are packed in a box. These bolts are randomly selected from a population with a mean weight of 1 ounce and a standard deviation of 0.01 ounces. Find the probability that the average weight of the bolts in the box is less than 1.0015 ounces. Ans. (0.9332)

Example 10: The scores on the ACT college entrance examination in a recent year had a mean of 18.6 and a standard deviation of 5.9. The average score of 76 students at Northside High who took the test was 20.4 What is the probability that the mean score for 76 students chosen randomly from all who took the test nationally is 20.4 or higher? Ans. (0.004) HW: 3, 4, 5-15 (odd) pp. 246-248