X n should be to μ mathematically the central limit

Info icon This preview shows pages 16–17. Sign up to view the full content.

View Full Document Right Arrow Icon
X n should be to μ . Mathematically, the central limit theorem says that n ( ¯ X n - μ ) d N (0 , σ 2 ). We cannot know the exact value of σ 2 merely by observing our sample X 1 , . . . , X n . But we can estimate it with the sample variance ˆ σ 2 n , defined as ˆ σ 2 n = 1 n n X i =1 ( X i - ¯ X n ) 2 . Some people prefer to define the sample variance with a divisor of n - 1 instead of n , but this makes no difference to us, since we will be looking for limiting results as n → ∞ . With a little algebra, we can see that ˆ σ 2 n = 1 n n X i =1 X 2 i - ¯ X 2 n . 16
Image of page 16

Info icon This preview has intentionally blurred sections. Sign up to view the full version.

View Full Document Right Arrow Icon
The law of large numbers tells us that 1 n n i =1 X 2 i p E ( X 2 ) as n → ∞ . It also tells us that ¯ X n p E ( X ), and so the continuous mapping theorem implies that - ¯ X 2 n p - E ( X ) 2 . Therefore, applying part 1 of Slutsky’s theorem, we find that ˆ σ 2 n p E ( X 2 ) - E ( X ) 2 = Var( X ) = σ 2 as n → ∞ . In other words, the sample variance ˆ σ 2 n provides a consistent estimate of the population variance σ 2 . Since the square root function is continuous, the continuous mapping theorem implies that ˆ σ n p σ . Combining this result with part 3 of Slutsky’s theorem and the fact that n ( ¯ X n - μ ) d N (0 , σ 2 ), we find that ¯ X n - μ ˆ σ n / n = n ( ¯ X n - μ ) ˆ σ n d N (0 , σ 2 ) σ = N (0 , 1) , provided that σ > 0 (which will be true unless every X i takes the same value). We have shown that the distribution of the ratio ¯ X n - μ ˆ σ n / n approximates the N (0 , 1) distribution in large samples. This fact can be used to test hypotheses about the value of μ , or to form confidence intervals for μ . Suppose we wish to test the hypothesis that μ = μ 0 . Here, μ is the true expected value of X , which we do not know, and μ 0 is a conjectured value of μ . We can calculate the t -statistic t n = ¯ X n - μ 0 ˆ σ n / n from our sample X 1 , . . . , X n . If the hypothesis μ = μ 0 is true, then our earlier argument shows that t n has the N (0 , 1) distribution in large samples. Therefore, it should lie between - 1 . 96 and 1 . 96 with probability approximately equal to 0 . 95. On the other hand, if μ 6 = μ 0 , we have no reason to believe that t n will lie in this range, and in fact it can be shown that it will be greater than 1.96 in absolute value with probability approaching one as n → ∞ . Therefore, if we calculate t n and find that it does not lie between -1.96 and 1.96, we reject the hypothesis that μ = μ 0 . If our hypothesis μ = μ 0 is true, there is an approximately 5% chance that we will incorrectly reject it. This is usually deemed an acceptable rate of error for applied work in the social sciences. The approximation ¯ X n - μ ˆ σ n / n N (0 , 1) can also be used to form large sample confidence intervals for the unknown quantity μ . A standard normal random vari- able lies between -1.96 and 1.96 with probability 0.95. Recalling the definition of convergence in distribution, we can see that lim n →∞ P - 1 . 96 ¯ X n - μ ˆ σ n / n 1 . 96 = 0 . 95 .
Image of page 17
This is the end of the preview. Sign up to access the rest of the document.

{[ snackBarMessage ]}

What students are saying

  • Left Quote Icon

    As a current student on this bumpy collegiate pathway, I stumbled upon Course Hero, where I can find study resources for nearly all my courses, get online help from tutors 24/7, and even share my old projects, papers, and lecture notes with other students.

    Student Picture

    Kiran Temple University Fox School of Business ‘17, Course Hero Intern

  • Left Quote Icon

    I cannot even describe how much Course Hero helped me this summer. It’s truly become something I can always rely on and help me. In the end, I was not only able to survive summer classes, but I was able to thrive thanks to Course Hero.

    Student Picture

    Dana University of Pennsylvania ‘17, Course Hero Intern

  • Left Quote Icon

    The ability to access any university’s resources through Course Hero proved invaluable in my case. I was behind on Tulane coursework and actually used UCLA’s materials to help me move forward and get everything together on time.

    Student Picture

    Jill Tulane University ‘16, Course Hero Intern