MATH 104 CHAPTER 5 page 1 NORMAL DISTRIBUTION We have examined discrete random variables, those random variables for which we can list the possible values. We will now look at continuous random variables. A continuous random variable has infinitely many values resulting from measured quantities such a the net weight of a package of frozen food, the amount of tar in a cigarette, the time to complete a task, etc. One example of a continuous distribution is the uniform distribution. The domain of a uniform random variable X consists of all values within some interval a x b. The area under the graph of a uniform distribution between two values is the probability that the random variable will take on a number between those values. p(x) 1 b a 1 4 total area = 1 x a b 2 6 For X, a Uniform Random Variable, with 2 x 6 : example1: P( 2 < x < 3 ) = example 2: P(3 < x < 5) = example 3; P(2 < x < 6 ) = example 4: P(x < 3) = >>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>> Among the many continuous distribution curves used in statistics, the most important is the normal distribution. One reason for the importance of the normal distribution is that it usefully models or describes the distributions of many random variables that arise in practice, such as the heights or weights of a group of people, the total annual sales of a firm, the measurement errors that arise in the performance of an experiment. The normal distribution is also important in that it provides a useful approximation to many other distributions. The graph of a normal distribution is called the normal curve and is bell-shaped. The important parameters of a normal distribution are the mean and the standard deviation, µ and σ. There is one and only one normal distribution with a given mean and a given standard deviation. In a normal distribution: - the mean, median and mode are equal - the distribution is bell-shaped and symmetric about the mean As for any graph of a continuous probability, there is a correspondence between area and probability. We find the area under normal curves by using the calculus tool of 1 integration of the formula f(x) = σ 2π e (1/2){(x µ )/σ }2. But, in practice we obtain the areas under normal curves in tables such as the ones in the front cover of your textbook.
MATH 104 CHAPTER 5 page 2 In order to use the standard normal probability tables on the front cover of your textbook, the normal random variable must be standardized by using the z-scores. data.value mean Recall: the z-score of a data value = stan dard.deviation = x µ = z. This z-score measures σ how far the data value is from the mean of the distribution. It is expressed as a number of standard deviations. (We can use the Empirical Rule to approximate the probabilities, but we can get far more accuracy by converting a data value to its z-score and then using the standard normal tables in front of the text. example: The height of adult females is N(63.6, 2.5), which reads the random variable is normally distributed with mean of 63.6 inches and standard deviation 2.5 inches. For the questions below, draw and label a bell-shaped curve and for parts b, c, and d. shade the area of interest. a) What is the z-score of a woman who is 5 feet-6 inches? b) What percentage of adult females are shorter than 5 feet-6 inches? c) What percentage of adult females are taller than 5 feet-6 inches? d) What percentage of adult females are between 5 feet and 5 feet-6 inches?
MATH 104 CHAPTER 5 page 3 NORMAL DISTRIBUTION FOR EACH EXAMPLE GIVE A SKETCH, WITH SHADING 1. Find the area under the standard normal curve that lies to the left of z = 1.05 1. 2. Find the z - score of the standard normal curve having area of 0.37 to its left. 2. 3. The United States Army requires that women s heights be between 58 inches and 70 inches. Assume that U.S. women have heights that are normally distributed with a mean of 63.6 inches and a standard deviation of 2.5 inches. Find the percentage of U.S. women satisfying that requirement. 3. 4. In human engineering and product design, it is often important to consider the weights of people so that airplanes and elevators aren t overloaded, etc. Assume that the population of men has normally distributed weights, with a mean of 173 pounds and standard deviation of 30 pounds. a) If a man is randomly selected, find the probability that his weight is greater than 225 pounds. a) b) If 36 different men are randomly selected from this population, find the probability that their mean weight is greater than 180 pounds. b)
MATH 104 CHAPTER 5 page 4 example 2 :Suppose that a population of disk drives whose service life is a normally distributed random variable with mean 760 hours and standard deviation of 140 hours. a) Find the probability that a randomly selected disk drive will have a life of 550 hours or less. b) What is the probability that a drive will fail before 1000 hours have passed? c) What proportion of the disk drives can be expected to exceed a life of 900 hours? d) Find the probability that a drive will last longer than 600 hours. e) What is the probability that a randomly selected disk drive will be one whose lifetime is between 700 and 800 hours? f) Find the point that is the 40th percentile of this distribution. g) Find the amount of time that has elapsed when only 2% of the population of disk drives are still working. h) Determine the points between which the middle 80% of the distribution does lie.
MATH 104 CHAPTER 5 page 5 SAMPLING DISTRIBUTIONS CENTRAL LIMIT THEOREM We can estimate the mean of a population, µ, (a parameter) by choosing a random sample from the population and calculating the mean from the sample, x, (a statistic). While there is very little chance that the sample mean and the population mean are identical, we expect them to be quite close. For the purpose of statistical inference, we need to measure how close the sample mean is likely to be to the population mean. Different samples will give us different estimates. Sampling error is the difference between the sample measure and the corresponding population measure. (ex. the difference between µ and x ) If we repeatedly draw samples and calculate the mean of each sample, these sample means will have a distribution that we call a sampling distribution. A sampling distribution of these means (distribution of the x 's ) is a distribution obtained by using the means computed from random samples of a specific size taken from a population. >>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>> CENTRAL LIMIT THEOREM: When you select simple random samples of size n from a population with mean µ and standard deviation σ, then the values of the samples means, the x 's, calculated from the samples have a mean equal to the mean of the population mean, µ, µ x = µ and have a standard deviation of σ n, σ x = σ n. If the sample size, n is relatively large,(ie: greater than or equal to 30), then the distribution of the sample means is normal or bell-shaped x s have a distribution of N( µ, σ n ) >>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>> Our goal is to find out how far our sample mean ( an x value) is from the true population mean, µ. So we wish to estimate our sampling error, which is the standard deviation of the sampling means, σ x = σ. This is called the standard error of the sample n means. (On a Minitab printout: SEMEAN is the standard error of the mean) example: A local survey of all churches finds that the mean number of members per church in the area is 458. Assume a standard deviation of 116.7. For all the possible samples of size 50: a) find the mean of all of the possible sample means, (the mean of the x 's) b) find the sampling error of the distribution of the sample means, ( σ x ) c) According to the Central Limit Theorem, what is the distribution of all of the possible sample means?
MATH 104 CHAPTER 5 page 6 PRACTICE 1. Find the following probabilities for the given z-scores; a) P(z<2.78) = b) P(-2.1<z<2.33) = c) P(z>2.85) = d) P(z< - 0.13) = 2. Find the z-score for which the a) area to its left is 0.86 z = b) area to its right is 0.24 z = 3. The amount of time that customers wait in line during peak hours at one bank is normally distributed with a mean of 16 minutes and a standard deviation of 2 minutes. The percentage of time that the waiting time lies between 16 and 18 minutes is 4. The mean annual income for adult women in one city is $28,520 and the standard deviation of the incomes is $5,190. The distribution of the incomes is skewed to the right. For samples of size 40, describe the sampling distribution of the mean. x has a distribution the mean of all the possible sample means, (from samples, size 40), is the standard deviation of all the possible sample means (sample size 40) is 5. The National Weather Service keeps records of snowfall in mountain ranges. Records indicate that in a certain range, the annual snowfall has a mean of 103 inches and a standard deviation of 16 inches. a)for a survey of 35 years, what is the probability that the mean snowfall of the 35 years is more than 108 inches? b) From a sample of 35 years, would it be likely that the mean of the sample equals 117? Why or why not?
MATH 104 CHAPTER 5 page 7 MINITAB ASSIGNMENT NAME Follow the directions for each part, using your results to answer the included questions as directed. PART I: 1. CALC>RANDOM>NORMAL generate 100 rows of data store into c1-c40 mean = 20 standard deviation = 3.5 OK 2. MANIP>STACK stack c1-c40 click on the selection column of current worksheet and enter c41 OK 3. name c41 STACKN (by typing the name at the top of the column) 4. CALC>ROW STATISTICS mean input variables: c1-c40 store into c42 5. name c42 XbarN 6. STAT>BASIC STATISTICS>DISPLAY DESCRIPTIVE STATISTICS variables: c41-c42 OK 7. GRAPH>HISTOGRAM under x, enter: c41 OK 8. FILE>PRINT GRAPH 9. GRAPH>HISTOGRAM under x, enter: c42 OK 10. FILE>PRINT GRAPH 11. retrieve graphs from printer 12. highlight ALL columns and hit the DELETE key PART II: Repeat steps 1-12 with the following changes to the expressions printed in bold: step 1: replace NORMAL with EXPONENTIAL and mean = 10 step 3: name c41 STACKE step 5: name c42 XbarE PART III: Repeat steps 1-12 with the following changes to the expressions printed in bold: step 1: replace NORMAL with UNIFORM lower endpoint = 20 upper endpoint = 60 step 3. name c41 STACKU step 5. name c42 XbarU PART IV: CLICK on the Session Window so that the word session is highlighted FILE>PRINT SESSION WINDOW ANSWER THE QUESTIONS ON THE ATTACHED SHEET
MATH 104 CHAPTER 5 page 8 MATH 104 MINITAB ASSIGNMENT #2 NAME PART I: 1. Give the mean for the population in part I 2. Give the standard deviation for the population in part 1 3. Give the mean of XbarN in part I 4. Give the standard deviation of XbarN in part I 5. According to the Central Limit Theorem, if every possible sample of size 30 is taken, what is the value of the mean of all the sample means? 6. According to the Central Limit Theorem, if every possible sample of size 30 is taken, what is the value of the standard deviation of all the sample means? PART II: 1. Given the mean for the population in part II 2. Give the mean of XbarE in part II 3. According to the Central Limit Theorem, if every possible sample of size 30 is taken, what is the value of the mean of all the sample means? 4. According to the Central Limit Theorem, if every possible sample of size 30 is taken, what is the value of the standard deviation of all the sample means? PART III: 1. Give the mean for the population in part III 2. Give the mean of XbarU in part III 3. According to the Central Limit Theorem, if every possible sample of size 30 is taken, what is the value of the mean of all the sample means? 4. According to the Central Limit Theorem, if every possible sample of size 30 is taken, what is the value of the standard deviation of all the sample means? YOU WILL TURN IN 8 STAPLED PAGES: 1.THIS PAGE, 2.THE SESSION WINDOW, 3-8. THE HISTOGRAMS (in the order in which you printed them)
MATH 104 CHAPTER 5 page 9 SECTION 5.6 THE NORMAL APPROXIMATION TO THE BINOMIAL DISTRIBUTION The Binomial probability formula can be used to compute probabilities of an event in a binomial distribution. When there are a large number of trials in a binomial experiment, the binomial probability can be difficult to use. Recall that in a binomial experiment, as the number of trials, n, increases, the probability becomes more nearly symmetric and bell shaped. n p 5 AND n q 5 the probability As a general rule of thumb: if distribution will be approximately symmetric and bell shaped. Because of this, we can approximate the binomial by the area under the normal curve. (If either n p 5 or n q 5 then the binomial distribution is too skewed for the normal curve to give accurate approximations,) So: If n p 5 AND n q 5, then the binomial random variable X is approximately normally distributed with µ x = np and σ x = n p q Correction for Continuity: When using a continuous random variable to approximate a discrete random variable, move 0.5 units to the left or right to include all possible values of x. EXAMPLES: Assume for 1-4 that n p 5 and n q 5 1. P(X BINOMIAL 10) P( X NORMAL ) 2. P(X BINOMIAL < 10) P(X NORMAL < ) 3. P( X BINOMIAL 10) P(X NORMAL ) 4. P( X BINOMIAL >10) P( X NORMAL > )
MATH 104 CHAPTER 5 page 10 SECTION 5.6 CONTINUED NAME 5. Suppose that X ~ BINOMIAL(n = 26, p = 0.3) Use the normal approximation to the binomial distribution to find P(X > 6) To use the Normal distribution to approximate the Binomial: first check the conditions: n p 5 and n q 5 find the mean and standard deviation: µ x = np = σ x = n p q = Correction for Continuity P(X binomial > 6) ( X normal ) Calculate the normal probability where z = x np npq or use your calculator nmcdf (LB, UB, µ = n p, σ = n p ( 1 p) ) answer = 6. For the information above to use the normal distribution to approximate the the binomial probability: a. P(X binomial 6) (X normal ) = b. P( X binomial < 9) (X normal ) = c. P(5 X binomial < 9) ( X normal ) = 7. For a binomial random variable with n = 50 and p = 65%, use the normal approximation to the binomial distribution to find each ROUND TO FOUR DECIMAL PLACES...check conditioins first a. P(X binomial 28) (X normal ) a. b. P(22 X binomial < 29) ( X normal ) b.