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Unformatted text preview: 467 2.763 29 1.311 1.699 2.045 2.462 2.756 30 1.310 1.697 2.042 2.457 2.750 40 1.303 1.684 2.021 2.423 2.704 60 1.296 1.671 2.000 2.390 2.660 120 1.289 1.658 1.980 2.358 2.617 ∞ 1.282 1.645 1.960 2.326 2.576 t 15 Inference using the t distribution l Strategies for constructing confidence intervals & conducting tests remain intact l What has changed is the distribution of our test statistic l What is a (100 α )% CI for the sample mean when X ~ N ( μ , σ 2 ) but σ 2 is unknown? freedom of degrees 1 with on distributi the using determined be will values critical but rules decision define to used be to statistic test the is Similarly for CI )% (100 the is 1 , 2 / n t n s X t n s t X n μ μ α α = ± 16 Final exam June 2005 Q3(b) l Assume that an investor has a portfolio of investments l From past evidence, annual returns can be treated as approximately normal l In a small sample of 9 investments, the sample mean=15% & (sample) standard deviation=11% l In question were not told but you needed to realize l CLT is inappropriate, but sampling from a normal population, so not needed Final exam June 2005 Q3(b)... l Consider the following questions l (i) Assume population standard deviation to be 12%. Suppose CI has a lower limit of 8.42%. What confidence level was used? l (ii) If use a higher confidence level what happens to the width of the CI? l (iii) Assume the population standard deviation is unknown. Find the 95% CI for the population mean return l (iv) Interpret the numerical result in (iii). 17 18 Final exam June 2005 Q3(b)… interval wider confidence of level in increase se (ii)Decrea CI 90% a used hence & % 10 05 . ) 645 . 1 ( 645 . 1 4 / ) 42 . 8 15 ( 3 12 15 8.42 or n be will bound lower & n is CI )100% (1 then 12 If (i) ) 1 , ( ~ and /n) , ( then ) , ( be returns let & 11 , 15 , 9 Know 2 / 2 / 2 / 2 / 2 2 ⇒ ⇒ = ⇒ = = = = ± = = ⇒ = = = α α σ σ α σ σ μ σ μ σ μ α α α α Z P z z z X z X N n X Z ~N X X~N s x n 19 Final exam June 2005 Q3(b)… return mean population the include would manner this in d constructe intervals the of 95% that expect would you returns of population this from drawn 9 size of samples repeated (iv)In ) 46 . 23 , 54 . 7 ( or 3 11 .306 2 15 n by given CI 95% & ~ (iii) of use unknown Assuming 1 , 2 / 1 ± ± ⇒ s t X t n s X t n n α μ σ 20 Final exam June 2005 Q3(b)… l How would you answer (iii) if you were not willing to assume normality for the underlying population? 21 Inferences about other parameters? l Have concentrated on inference for the population mean l Concepts & procedures can be applied to testing & creating CI’s for many other situations, including l Population proportion l Regression parameters l Difference in means l Population variance (first considered now, other 3 will be treated later in course) 22...
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 DenzilGFiebig
 Normal Distribution

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