STATISTICAL METHODS I
STATISTICAL METHODS I STAT 515
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This 12 page Class Notes was uploaded by Shane Marks on Monday October 26, 2015. The Class Notes belongs to STAT 515 at University of South Carolina - Columbia taught by Xi Huang in Fall. Since its upload, it has received 54 views. For similar materials see /class/229646/stat-515-university-of-south-carolina-columbia in Statistics at University of South Carolina - Columbia.
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Date Created: 10/26/15
STAT 515 Chapter 6 Sampling Distributions Definition Parameter a number that characterizes a population example population mean u it s typically unknown Statistic a number that characterizes a sample example sample mean X we can calculate it from our sample data We use the sample mean X to estimate the population mean u Suppose we take a sample and calculate X Will X equal 4 Will X be close to 4 Suppose we take another sample and get another X Will it be same as rst X Will it be close to first X 0 What if we took many repeated samples of the same size from the same population and each time calculated the sample mean What would that set of X values look like The sampling distribution of a statistic is the distribution of values of the statistic in all possible samples of the same size from the same population Consider the sampling distribution of the sample mean X when we take samples of size n from a population with mean u and variance 0392 Picture The sampling distribution of X has mean u and standard deviation 0 J Notation Point Estimator A statistic which is a single number meant to estimate a parameter It would be nice if the average value of the estimator over repeated sampling equaled the target parameter An estimator is called unbiased if the mean of its sampling distribution is equal to the parameter being estimated Examples Another nice property of an estimator we want the spread of its sampling distribution to be as small as possible The standard deviation of a statistic s sampling distribution is called the standard error of the statistic The standard error of the sample mean X is 0N Note As the sample size gets larger the spread of the sampling distribution gets smaller When the sample size is large the sample mean varies less across samples Evaluating an estimator 1 Is it unbiased 2 Does it have a small standard error Central Limit Theorem We have determined the center and the spread of the sampling distribution of X What is the shape of its sampling distribution Case I If the distribution of the original data is normal the sampling distribution of X is normal This is true no matter what the sample size is Case 11 Central Limit Theorem If we take a random sample of size n from any population with mean u and standard deviation 039 the sampling distribution of X is approximately normal if the sample size is large How large does 11 have to be Our rule of thumb If n 2 30 we can apply the CLT result Pictures As 11 gets larger the closer the sampling distribution looks to a normal distribution Why is the CLT important Because when X is approximately normally distributed we can answer probability questions about the sample mean Standardizing values of X If X is normal with mean u and standard deviation 0 J then Y 039 J has a standard normal distribution Z Example Suppose we re studying the failure time at high stress of a certain engine part The failure times have a mean of 14 hours and a standard deviation of 09 hours If our sample size is 40 engine parts then what is the sampling distribution of the sample mean What is the probability that the sample mean will be greater than 15 Example Suppose lawyers salaries have a mean of 90000 and a standard deviation of 30000 highly skewed Given a sample of lawyers can we find the probability the sample mean is less than 100000 ifn5 Ifn30 Other Sampling Distributions In practice the population standard deviation 039 is typically unknown We estimate 039 with s I But the quantity S J no longer has a standard normal distribution Its sampling distribution is as follows 0 If the data come from a normal population then the T X statistic SAM has a tdistribution Student s t with n 1 degrees of freedom the parameter of the tdistribution The tdistribution resembles the standard normal symmetric moundshaped centered at zero but it is more spread out o The fewer the degrees of freedom the more spread out the tdistribution is 0 As the df increase the tdistribution gets closer to the standard normal Picture Table VI gives values of the tdistribution with specific areas to the right of these values Verify In tdistribution with 3 df area to the right of is 025 Notation For 3 df t025 In t with 14 df area to the right of is 05 In t with 25 df area to the right of is 999 The x2 Chisquare Distribution Suppose our sample of size 11 comes from a normal population with mean u and standard deviation 039 n ls2 Then 02 has a 6 distribution with n 1 degrees of freedom 0 The x2 distribution takes on positive values 0 It is skewed to the right 0 It is less skewed for higher degrees of freedom 0 The mean of a x2 distribution with n 1 degrees of freedom is n 1 and the variance is 2n 1 Fact If we add the squares of n independent standard normal rv s the resulting sum has a x2 distribution n ls2 Note that 2 O We sacrifice one df by estimating u with X so it is x2n1 Table VII gives values of a x2 rv with speci c areas to the right of those values Examples For x2 with 6 df area to the right of is 90 For x2 with 6 df area to the right of is 05 For x2 with 80 df area to the right of is 10 The F Distribution 2 anil n1 The quantity 1271 n2 1 where the two 6 rv s are independent has an Fdistribution with 111 1 numerator degrees of freedom and n2 1 denominator degrees of freedom So if we have samples of sizes 111 and 112 from two normal populations note has an Fdistribution with 111 1 n2 1 df Table VIII gives values of F rv with area 10 to the right Table IX gives values of F rv with area 05 to the right Table X gives values of F rv with area 025 to the right Table XI gives values of F rv with area 01 to the right Verify For F with 3 9 df 281 has area 010 to right For F with 15 13 df 382 has area 001 to right 0 These sampling distributions will be important in many inferential procedures we will learn
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