chapter10 - Introduction to Probability and Statistics...

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1 Introduction to Probability and Statistics Thirteenth Edition Chapter 10 Inference from Small Samples Introduction • When the sample size is small, the estimation and testing procedures of Chapter 8 are not appropriate. • There are equivalent small sample test and estimation procedures for m , the mean of a normal population m 1 -m 2 , the difference between two population means s 2 , the variance of a normal population The ratio of two population variances. The Sampling Distribution of the Sample Mean • When we take a sample from a normal population, the sample mean has a normal distribution for any sample size n, and • has a standard normal distribution. • But if s is unknown, and we must use s to estimate it, the resulting statistic is not normal . / x z n m s - is not normal! / x sn - x Student’s t Distribution Fortunately, this statistic does have a sampling distribution that is well known to statisticians, called the Student’s t distribution, with n -1 degrees of freedom. / x t - •We can use this distribution to create estimation testing procedures for the population mean m .
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2 Properties of Student’s t Shape depends on the sample size n or the degrees of freedom, n -1. As n increases the shapes of the t and z distributions become almost identical. Mound-shaped and symmetric about 0. More variable than z , with ―heavier tails‖ Using the t -Table Table 4 gives the values of t that cut off certain critical values in the tail of the t distribution. Index df and the appropriate tail area a to find t a , the value of t with area a to its right. For a random sample of size n = 10, find a value of t that cuts off .025 in the right tail. Row = df = n –1 = 9 t .025 = 2.262 Column subscript = a = .025 Small Sample Inference for a Population Mean m The basic procedures are the same as those used for large samples. For a test of hypothesis: . 1 on with distributi - t a on based region rejection a or values - using / statistic test the using tailed or two one : H versus : H Test 0 a 0 0 - - n df p n s x t m For a 100(1 -a )% confidence interval for the population mean m: . 1 on with distributi - t a of tail in the /2 area off cuts that of value the is where 2 / 2 / - n df t t n s t x a Small Sample Inference for a Population Mean m
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3 Example A sprinkler system is designed so that the average time for the sprinklers to activate after being turned on is no more than 15 seconds. A test of 5 systems gave the following times: 17, 31, 12, 17, 13, 25 Is the system working as specified? Test using a = .05. specified) as ng (not worki 15 : H as (working 15 : H a 0 m Example Data: 17, 31, 12, 17, 13, 25 First, calculate the sample mean and standard deviation, using your calculator or the formulas in Chapter 2. 387 . 7 5 6 115 2477 1 ) ( 167 . 19 6 115 2 2 2 - - - n n x x s n x x i Example Data: 17, 31, 12, 17, 13, 25 Calculate the test statistic and find the rejection region for a =.05.
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This note was uploaded on 09/19/2011 for the course MTH 1250C taught by Professor Any during the Fall '08 term at St. Johns Duplicate.

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chapter10 - Introduction to Probability and Statistics...

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