Freedom of degrees 1 with on distributi the using

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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 - = ± -
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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
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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
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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 , 0 ( ~ 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
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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
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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?
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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)
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22 Estimating the population proportion l Recall binomial rv X l Number of successes in n trials l Know that E( X )= np & Var( X )= np(1-p) l Now consider the rv X / n l This is simply the sample proportion of successes l It is an unbiased estimator of the population proportion l Why? l It has a variance of p(1-p)/n l Why?
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23 Estimating the population proportion… l Also recall that for large n, X is approximately normal l Thus the sample proportion is also approximately normal for large n using the CLT l Approximate sampling distribution follows immediately l As do confidence intervals & hypothesis tests ) 1 , 0 ( ~ / ) 1 ( ˆ and ) 1 ( , ˆ then ˆ Let N n p p p P Z n p p p ~N P n X P - - = - =
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