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Unformatted text preview: BUS12005 1 Part 5 More Inference about a Population 1. Inference about a Normal Population Mean: 2 Unknown 2. Inference about a Population Proportion Based on Large Samples Section 5.1 Inference about a Normal Population Mean: 2 Unknown Assumption The population follows N( , 2 ). Let X be the sample mean of a sample taken from the population, S be the sample standard deviation and n be the sample size. It can be proved that n S X / follows the t distribution with n 1 degrees of freedom . The probability density function of t ( n ) (the t distribution with n degrees of freedom where BUS12005 2 n is a positive integer) is f( t ) = 2 1 2 1 2 2 1 + + + n n t n n n where ( y ) = + u e u u y d 1 for y > 0. The graph below shows the probability density functions of N(0, 1) and some t ( n ). Interval estimation The 100(1 )% confidence interval for the 3 2 1 1 2 3 0.1 0.2 0.3 0.4 t (1) t (5) t (20) N(0, 1) BUS12005 3 population mean is + n s t x n s t x n n 1 , 2 / 1 , 2 / , where the value t , n can be obtained from the t distribution table . Comparing with the case with known , here we use s instead of and t / 2, n 1 instead of z / 2 . Density function of the t distribution with n degrees of freedom Example 1 A paint manufacturer wants to determine the average drying time of a new brand of interior wall paint. If for 12 test areas of equal size he obtained a mean drying time of 66.3 minutes and a standard deviation of 8.4 minutes, construct a 95% confidence interval for the true population mean t , n Area = BUS12005 4 assuming normality. [Solution] n = 12, x = 66.3, s = 8.4, = 1 0.95 = 0.05 and t / 2, n 1 = t 0.025,11 = 2.201. The 95% confidence interval for is + 12 4 . 8 201 . 2 3 . 66 , 12 4 . 8 201 . 2 3 . 66 , that is, (61.0, 71.6). Hypothesis test When the null hypothesis is = , or , the test statistic is chosen to be n S X / ....
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This note was uploaded on 11/03/2011 for the course ECON 101 taught by Professor Wood during the Spring '07 term at University of California, Berkeley.
 Spring '07
 Wood

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