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Unformatted text preview: BUS12005 1 Part 5 More Inference about a Population 1. Inference about a Normal Population Mean: σ 2 Unknown 2. Inference about a Population Proportion Based on Large Samples Section 5.1 Inference about a Normal Population Mean: σ 2 Unknown Assumption The population follows N( µ , σ 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 / µ − 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 + − ⎟ ⎠ ⎞ ⎜ ⎝ ⎛ + ⎟ ⎠ ⎞ ⎜ ⎝ ⎛ Γ ⎟ ⎠ ⎞ ⎜ ⎝ ⎛ + Γ n n t n n n π where Γ ( 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 − α )% 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 µ is ⎟ ⎠ ⎞ ⎜ ⎝ ⎛ + − − − n s t x n s t x n n 1 , 2 / 1 , 2 / , α α where the value t α , n can be obtained from the t distribution table . Comparing with the case with known σ , here we use s instead of σ and t α / 2, n − 1 instead of z α / 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 α , n Area = α BUS12005 4 assuming normality. [Solution] n = 12, x = 66.3, s = 8.4, α = 1 − 0.95 = 0.05 and t α / 2, n − 1 = t 0.025,11 = 2.201. The 95% confidence interval for µ 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 µ = µ , µ ≥ µ or µ ≤ µ , the test statistic is chosen to be n S X / µ − ....
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 Spring '07
 Wood
 Statistics, Normal Distribution, Statistical hypothesis testing, µ

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