University of California, Los AngelesDepartment of StatisticsStatistics 13 Instructor: Nicolas ChristouThe central limit theoremThe distribution of the sample proportionThe distribution of the sample meanThe distribution of the sample proportion:• First: Population proportion, p:• Sample proportion, ˆp.• Very important: The sample proportion, ˆp, follows the normal distribution with meanp, and standard deviationqp(1−p)n, where p is the population proportion, and n is thesample size.Requirements: large n, and independent sample values.• What does the above statement mean?1• We can write:ˆp ∼ Np,sp(1 − p)nThereforeZ =ˆp − pqp(1−p)n)• How can we use the above statement? Here is an example:Suppose the population proportion of Democtrats among UCLA students is 55%. Findthe probability that the sample proportion of 200 students randomly selected willexceed 60%.• Important:The population proportion p is a parameter, it is a fixed number between 0 and 1. Itis not random!The sample proportion ˆp is not fixed. It is random!2The distribution of the sample mean:• First: Population mean, µ:• Sample mean,¯X:• Very important: The sample mean,¯X, follows the normal distribution with mean µ,and standard deviationσ√n, where µ and σ are the mean and standard deviation of thepopulation from where the sample of size n is selected.Requirements: large n, and independent sample values.• What does the above statement mean?3• We can write:¯X ∼ N µ,σ√n!ThereforeZ =¯X − µσ√n• As a result of the previous statement the following is also true (and very useful):T ∼ Nnµ, σ√nThereforeZ =T − nµσ√n• How can we use the above statement? Here is an example:A large freight elevator can transport a maximum of 9800 pounds. Suppose a load ofcargo containing 49 boxes must be transported via the elevator. Experience has shownthat the weight of boxes of this type of cargo follows a distribution with mean µ = 205pounds and standard deviation σ = 15 pounds. Based on this information, what isthe probability that all 49 boxes can be safely loaded onto the freight elevator andtransported?• Important:The population mean µ is a parameter, it is a fixed number. It is not random!The sample mean¯X is not fixed. It is random!4Here are two simulation experiments from the SOCR sample mean experiment:A. Population N (4, 1.5). Sample size n = 16. We observe that because the populationfrom where the samples are selected is normal even a small sample (here n = 16) willproduce a bell-shaped distribution for the sample mean¯X.B. Population exp(λ = 1). Sample size n = 100:5Distribution of the sample mean - Sampling from normal distributionIf we sample from normal distribution N(µ, σ) then¯X follows exactly the normal distributionwith mean µ and standard deviationσ√nregardless of the sample size n. In the next figurewe see the effect of the sample size on the shape of the distribution of¯X. The first figureis the N(5, 2) distribution. The second figure represents the distribution of¯X when n = 4.The third figure represents the distribution of¯X when n = 16.xf((x))−3 −1 1 3 5 7 9 11 130.0 0.2 0.4 0.6 0.8N((5,, 2))xf((x))1 2 3 4 5 6 7 8 90.0 0.2 0.4 0.6 0.8N((5,, 24))xf((x))3 4 5 6 70.0 0.2 0.4 0.6 0.8N((5,, 216))6Sampling distributions - ExamplesExample 1A large freight elevator can transport a maximum of 9800 pounds. Suppose a load of cargo containing 49 boxes must be transported viathe elevator. Experience has shown that the weight of boxes of this type of cargo follows a distribution with mean µ = 205 pounds andstandard deviation σ = 15 pounds. Based on this information, what is the probability that all 49 boxes can be safely loaded onto thefreight elevator and transported?Example 2From past experience, it is known that the number of tickets purchased by a student standing in line at the ticket window for the footballmatch of UCLA against USC follows a distribution that has mean µ = 2.4 and standard deviation σ = 2.0. Suppose that few hoursbefore the start of one of these matches there are 100 eager students standing in line to purchase tickets. If only 250 tickets remain, whatis the probability that all 100 students will be able to purchase the tickets they desire?Example 3Suppose that you have a sample of 100 values from a population with mean µ = 500 and with standard deviation σ = 80.a. What is the probability that the sample mean will be in the interval (490, 510)?b. Give an interval that covers the middle 95% of the distribution of the sample mean.Example 4The amount of mineral water consumed by a person per day on the job is normally distributed with mean 19 ounces and standarddeviation 5 ounces. A company supplies its employees with 2000 ounces of mineral water daily. The company has 100 employees.a. Find the probability that the mineral water supplied by the company will not satisfy the water demanded by its employees.b. Find the probability that in the next 4 days the company will not satisfy the water demanded by its employees on at least 1 ofthese 4 days. Assume that the amount of mineral water consumed by the employees of the company is independent from day today.c. Find the probability that during the next year (365 days) the company will not satisfy the water demanded by its employees onmore than 15 days.Example 5Supply responses true or false with an explanation to each of the following:a. The probability that the average of 20 values will be within 0.4 standard deviations of the population mean exceeds the probabilitythat the average of 40 values will be within 0.4 standard deviations of the population mean.b. P (¯X > 4) is larger than P (X > 4) if X ∼ N (8, σ).c. If¯X is the average of n values sampled from a normal distribution with mean µ and if c is any positive number, then P (µ − c ≤¯X ≤ µ + c) decreases as n gets large.Example 6An insurance company wants to audit health insurance claims in its very large database of transactions. In a quick attempt to assess thelevel of overstatement of this database, the insurance company selects at random 400 items from the database (each item represents adollar amount). Suppose that the population mean overstatement of the entire database is $8, with population standard deviation $20.a. Find the probability that the sample mean of the 400 would be less than $6.50.b. The population from where the sample of 400 was selected does not follow the
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