The-Normal-Distributiondocx正常distributiondocx

1、.The Normal DistributionThis module introduces one of the most important distributions in statistics: the normal distribution.Learning ObjectivesAfter completion of this module the student will be able to· recognize the normal distribution· recognize the standard normal distribution·

2、perform calculations with the normal distribution using Excel· convert a normal distribution to a standard normal distributionKnowledge and Skills· normal distribution· standard normal distributionPrerequisites· sample mean, average· sample standard deviation· sample va

3、riance· percentileThis module relies heavily on Collaborative Statistics: Chapter 6 or online /content/m16979/latest/ Guide to Chapter 6Section 6.1 Read Subsections 6.1.1 and 6.1.2Length at BirthThe U.S. Department of Health of Human Services (HHS) is a U.S. government agency with

4、responsibility to “protect the health of all Americans and providing essential human services” (/about/). One of HHSs agencies is the Centers for Disease Control and Prevention (CDC), which oversees a number of coordinating centers/offices and the National Institutes of Health inclu

5、ding the Coordinating Center for Health Information and Service (CCHIS). The National Center for Health Statistics (NCHS) is an office within the CCHIS. According to NCHSs website, its mission “is to provide statistical information that will guide actions and policies to improve the health of the Am

6、erican people. As the Nation's principal health statistics agency, NCHS leads the way with accurate, relevant, and timely data”( /nchs/). Data on the health and nutritional status of adults and children in the U.S. is collected by the National Health and Nutrition Examination Su

7、rvey (NHANES). We will be looking at one of NHANES data sets, namely the growth curves for infants. The data files can be found at /growthcharts/html_charts/lenageinf.htm. Growth charts that provide percentiles for length and weight at various ages are important screening tool to mo

8、nitor the growth of infants and assess their nutritional status and health risks caused by, for instance, obesity. The following data Source: /growthcharts/html_charts/lenageinf.htm lists selected percentiles for height (cm) for boys at 0 months:Percentile3510255075909597Height cm44

9、.9345.5746.5548.1949.9951.7753.3654.3154.92The distribution of the length follows a certain pattern that is described by the normal distribution.Subsection 6.1.3We say that a random variable is normally distributed with mean and standard deviation if the probability density function is given byBecau

10、se of its importance in statistics, it is worthwhile to memorize the form of the density function. The cumulative distribution function needs to be calculated by a computer or looked up in tables.Figure 1: The probability density function of a normal distribution with mean 0 and standard deviation 1

11、.The probability density function of a normal distribution with mean 0 and standard deviation 1 is displayed in Figure 1. Note the bell-shaped curve. EXCEL has a function that returns the cumulative distribution function and the density function for a normal distribution with a specified mean and st

12、andard deviation. The syntax isNORMDIST(x,mean,standard_dev,cumulative)Where x is the value at which you want to evaluate the distribution, mean is the mean, standard_dev is the standard deviation, and cumulative is a logical value (TRUE for the cumulative distribution function and FALSE for the pro

13、bability density function). In-class Activity 1(a) The height for boys at age 0 is normally distributed with mean 49.99cm and standard deviation 2.66cm. Use EXCEL to confirm the percentiles given in the table above.Instruction: To determine the percentile of the value 44.93cm, you need to calculate

14、the probability that the height is less than or equal to the value 44.93cm. The following function in EXCEL calculates this value=NORMDIST(44.93,49.99,2.66, TRUE)Rounded to two decimal places, the answer is 0.03, which is the 3rd percentile, thus confirming the result in the table. Repeat this for t

15、he other values given in the table.(b) Plot in EXCEL the cumulative distribution function and the probability density function for a normally distributed random variable with mean 49.99 and standard deviation 2.66 for .Section 6.2Read Section 6.2.The normal distribution is described by two parameter

16、s, the mean and the standard deviation . There are thus infinitely many such distribution. However, it turns out that they are all related. Namely, if a random variable is normally distributed with mean and standard deviation , we can define a new random variable, Z, as followsThis random variable i

17、s then normally distributed with mean 0 and standard deviation 1. Section 6.3Homework: Read Section 6.3 and work through all examples and problems.Section 6.4Homework: Read Section 6.4Section 6.5Homework: Read Section 6.5 and work through all examples and problems. Below are solutions to two of the

18、problems using EXCEL functions.Problem 1: Find when X is normally distributed with mean 63 and standard deviation 5: .Solution: To find in EXCEL, enter “=1-NORMDIST(65,63,5,TRUE)”. You should get “0.344578” as your answer.EXCEL has a function that returns the inverse of the cumulative distribution f

19、unction of a normal distribution with mean and standard deviation . The syntax isNORMINV(probability, mean, standard_dev)This function is useful when calculating percentiles as in the following problem.Problem 3: Find the 90th percentile of a random variable with distribution .Solution: In EXCEL, en

20、ter “=NORMINV(0.9,63,5)”. You should get “69.40776” as your answer. Homework ProjectThe following data gives the length-at-birth for girls:Percentile3510255075909597Height cm45.0945.5846.3447.6849.2951.0252.7053.7754.50Assume that the height-at-birth is normally distributed. Find the mean and the standard de