The Central Limit Theorem

Transcription

1 Objective Ue the central limit theorem to olve problem involving ample mean for large ample. The Central Limit Theorem In addition to knowing how individual data value vary about the mean for a population, tatitician are intereted in knowing how the mean of ample of the ame ize taken from the ame population vary about the population mean. Ditribution of Sample Mean Suppoe a reearcher elect a ample of 30 adult male and find the mean of the meaure of the triglyceride level for the ample ubject to be 187 milligram/deciliter. Then uppoe a econd ample i elected, and the mean of that ample i found to be 192 milligram/deciliter. Continue the proce for 100 ample. What happen then i that the mean become a random variable, and the ample mean 187, 192, 184,..., 196 contitute a ampling ditribution of ample mean. A ampling ditribution of ample mean i a ditribution uing the mean computed from all poible random ample of a pecific ize taken from a population. If the ample are randomly elected with replacement, the ample mean, for the mot part, will be omewhat different from the population mean m. Thee difference are caued by ampling error. Sampling error i the difference between the ample meaure and the correponding population meaure due to the fact that the ample i not a perfect repreentation of the population. When all poible ample of a pecific ize are elected with replacement from a population, the ditribution of the ample mean for a variable ha two important propertie, which are explained next. Propertie of the Ditribution of Sample Mean 1. The mean of the ample mean will be the ame a the population mean. 2. The tandard deviation of the ample mean will be maller than the tandard deviation of the population, and it will be equal to the population tandard deviation divided by the quare root of the ample ize. The following example illutrate thee two propertie. Suppoe a profeor gave an 8-point quiz to a mall cla of four tudent. The reult of the quiz were 2, 6, 4, and 8. For the ake of dicuion, aume that the four tudent contitute the population. The mean of the population i m The tandard deviation of the population i The graph of the original ditribution i hown in Figure Thi i called a uniform ditribution.

2 Chapter 6 The Normal Ditribution Figure 6 29 Ditribution of Quiz Score Hitorical Note Two mathematician who contributed to the development of the central limit theorem were Abraham DeMoivre ( ) and Pierre Simon Laplace ( ). DeMoivre wa once jailed for hi religiou belief. After hi releae, DeMoivre made a living by conulting on the mathematic of gambling and inurance. He wrote two book, Annuitie Upon Live and The Doctrine of Chance. Laplace held a government poition under Napoleon and later under Loui XVIII. He once computed the probability of the un riing to be 18,226,214/ 18,226,215. Frequency Now, if all ample of ize 2 are taken with replacement and the mean of each ample i found, the ditribution i a hown. Sample Mean Sample Mean 2, 2 2 6, 2 4 2, 4 3 6, 4 5 2, 6 4 6, 6 6 2, 8 5 6, 8 7 4, 2 3 8, 2 5 4, 4 4 8, 4 6 4, 6 5 8, 6 7 4, 8 6 8, 8 8 A frequency ditribution of ample mean i a follow. f For the data from the example jut dicued, Figure 6 30 how the graph of the ample mean. The hitogram appear to be approximately normal. The mean of the ample mean, denoted by, i m X _ X Score m X Figure 6 30 Ditribution of Sample Mean Frequency Sample mean

3 which i the ame a the population mean. Hence, mx _ m The tandard deviation of ample mean, denoted by X _ which i the ame a the population tandard deviation, divided by 2: X _ (Note: Rounding rule were not ued here in order to how that the anwer coincide.) In ummary, if all poible ample of ize n are taken with replacement from the ame population, the mean of the ample mean, denoted by, equal the population mx _ mean m; and the tandard deviation of the ample mean, denoted by, equal n. X _ The tandard deviation of the ample mean i called the tandard error of the mean. Hence, X _ n A third property of the ampling ditribution of ample mean pertain to the hape of the ditribution and i explained by the central limit theorem. X _, i The Central Limit Theorem A the ample ize n increae without limit, the hape of the ditribution of the ample mean taken with replacement from a population with mean m and tandard deviation will approach a normal ditribution. A previouly hown, thi ditribution will have a mean m and a tandard deviation n. If the ample ize i ufficiently large, the central limit theorem can be ued to anwer quetion about ample mean in the ame manner that a normal ditribution can be ued to anwer quetion about individual value. The only difference i that a new formula mut be ued for the z value. It i n Notice that X i the ample mean, and the denominator mut be adjuted ince mean are being ued intead of individual data value. The denominator i the tandard deviation of the ample mean. If a large number of ample of a given ize are elected from a normally ditributed population, or if a large number of ample of a given ize that i greater than or equal to 30 are elected from a population that i not normally ditributed, and the ample mean are computed, then the ditribution of ample mean will look like the one hown in Figure Their percentage indicate the area of the region. It important to remember two thing when you ue the central limit theorem: 1. When the original variable i normally ditributed, the ditribution of the ample mean will be normally ditributed, for any ample ize n. 2. When the ditribution of the original variable might not be normal, a ample ize of 30 or more i needed to ue a normal ditribution to approximate the ditribution of the ample mean. The larger the ample, the better the approximation will be.

4 Figure 6 31 Ditribution of Sample Mean for a Large Number of Sample 34.13% 34.13% 2.28% 13.59% 13.59% 2.28% 3 X 2 X 1 X + 1 X + 2 X + 3 X Example 6 13 through 6 15 how how the tandard normal ditribution can be ued to anwer quetion about ample mean. Example 6 13 Hour That Children Watch Televiion A. C. Neilen reported that children between the age of 2 and 5 watch an average of 25 hour of televiion per week. Aume the variable i normally ditributed and the tandard deviation i 3 hour. If 20 children between the age of 2 and 5 are randomly elected, find the probability that the mean of the number of hour they watch televiion will be greater than 26.3 hour. Source: Michael D. Shook and Robert L. Shook, The Book of Odd. Solution Since the variable i approximately normally ditributed, the ditribution of ample mean will be approximately normal, with a mean of 25. The tandard deviation of the ample mean i X _ n The ditribution of the mean i hown in Figure 6 32, with the appropriate area haded. Figure 6 32 Ditribution of the Mean for Example The z value i n The area to the right of 1.94 i , or 2.62%. One can conclude that the probability of obtaining a ample mean larger than 26.3 hour i 2.62% [i.e., P( X 26.3) 2.62%].

5 Example 6 14 The average age of a vehicle regitered in the United State i 8 year, or 96 month. Aume the tandard deviation i 16 month. If a random ample of 36 vehicle i elected, find the probability that the mean of their age i between 90 and 100 month. Source: Harper Index. Solution Since the ample i 30 or larger, the normality aumption i not neceary. The deired area i hown in Figure Figure 6 33 Area Under a Normal Curve for Example The two z value are z z To find the area between the two z value of 2.25 and 1.50, look up the correponding area in Table E and ubtract one from the other. The area for z 2.25 i , and the area for z 1.50 i Hence the area between the two value i , or 92.1%. Hence, the probability of obtaining a ample mean between 90 and 100 month i 92.1%; that i, P(90 100) 92.1%. X Student ometime have difficulty deciding whether to ue The formula z X m n z X m n hould be ued to gain information about a ample mean, a hown in thi ection. The formula or i ued to gain information about an individual data value obtained from the population. Notice that the firt formula contain X, the ymbol for the ample mean, while the econd formula contain X, the ymbol for an individual data value. Example 6 15 illutrate the ue of the two formula.

6 Example 6 15 Meat Conumption The average number of pound of meat that a peron conume per year i pound. Aume that the tandard deviation i 25 pound and the ditribution i approximately normal. Source: Michael D. Shook and Robert L. Shook, The Book of Odd. a. Find the probability that a peron elected at random conume le than 224 pound per year. b. If a ample of 40 individual i elected, find the probability that the mean of the ample will be le than 224 pound per year. Solution a. Since the quetion ak about an individual peron, the formula z (X m) i ued. The ditribution i hown in Figure Figure 6 34 Area Under a Normal Curve for Part a of Example 6 15 The z value i Ditribution of individual data value for the population The area to the left of z 0.22 i Hence, the probability of electing an individual who conume le than 224 pound of meat per year i , or 58.71% [i.e., P(X 224) ]. b. Since the quetion concern the mean of a ample with a ize of 40, the formula z ( X m) ( n) i ued. The area i hown in Figure Figure 6 35 Area Under a Normal Curve for Part b of Example 6 15 The z value i Ditribution of mean for all ample of ize 40 taken from the population z X m n The area to the left of z 1.42 i

7 Hence, the probability that the mean of a ample of 40 individual i le than 224 pound per year i , or 92.22%. That i, P( X 224) Comparing the two probabilitie, you can ee that the probability of electing an individual who conume le than 224 pound of meat per year i 58.71%, but the probability of electing a ample of 40 people with a mean conumption of meat that i le than 224 pound per year i 92.22%. Thi rather large difference i due to the fact that the ditribution of ample mean i much le variable than the ditribution of individual data value. (Note: An individual peron i the equivalent of aying n 1.) Intereting Fact The bubonic plague killed more than 25 million people in Europe between 1347 and Finite Population Correction Factor (Optional) The formula for the tandard error of the mean n i accurate when the ample are drawn with replacement or are drawn without replacement from a very large or infinite population. Since ampling with replacement i for the mot part unrealitic, a correction factor i neceary for computing the tandard error of the mean for ample drawn without replacement from a finite population. Compute the correction factor by uing the expreion N n N 1 where N i the population ize and n i the ample ize. Thi correction factor i neceary if relatively large ample are taken from a mall population, becaue the ample mean will then more accurately etimate the population mean and there will be le error in the etimation. Therefore, the tandard error of the mean mut be multiplied by the correction factor to adjut for large ample taken from a mall population. That i, X _ n N n N 1 Finally, the formula for the z value become z X m n N n N 1 When the population i large and the ample i mall, the correction factor i generally not ued, ince it will be very cloe to The formula and their ue are ummarized in Table 6 1. Table 6 1 Formula z X m n Summary of Formula and Their Ue Ue Ued to gain information about an individual data value when the variable i normally ditributed. Ued to gain information when applying the central limit theorem about a ample mean when the variable i normally ditributed or when the ample ize i 30 or more.