{[ promptMessage ]}

Bookmark it

{[ promptMessage ]}

Nov 5th Sampling Distribution of Sample Variance point and interval estimate-1

Nov 5th Sampling Distribution of Sample Variance point and interval estimate-1

This preview shows pages 1–4. Sign up to view the full content.

Sampling Distribution of Sample Variance Consider a random sample of n observations drawn from a population with unknown mean µ and unknown variance σ 2 , when x 1 , x 2 , …,x n are sample members 𝑠𝑠 2 = 1 ( 𝑛𝑛 − 1) ( 𝑥𝑥 𝑖𝑖 − 𝑥𝑥̅ ) 2 𝑛𝑛 𝑖𝑖=1 is called sample variance and the square root, s , is called sample standard deviation. Different random samples have different sample variance. The Expected value of the sample variance is the population variance 𝐸𝐸 [ 𝑠𝑠 2 ] = 𝜎𝜎 2 If we assume the underlying population distribution is normal, then it can be shown that the sample variance and the population variance are related through a probability distribution known as chi-square distribution

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

View Full Document
Chi-square distribution of sample and population variance using random sample of size n observations from a normally distributed population whose variance is 𝜎𝜎 2 and resulting sampling variance is 𝑠𝑠 2 , it can be shown that 𝛸𝛸 ( 𝑛𝑛−1 ) 2 = ( 𝑛𝑛 − 1) 𝑠𝑠 2 𝜎𝜎 2 = ( 𝑥𝑥 𝑖𝑖 − 𝑥𝑥̅ ) 2 𝜎𝜎 2 had a distribution known as chi-square ( 𝛸𝛸 ( 𝑛𝑛−1 ) 2 ) with (n – 1) degrees of freedom. The distribution is asymmetric and defined for only positive values since variance are all positive values. We can characterize a particular member of the family of the chi-square distributions by a single parameter referred to as degree of freedom, denoted as v . A chi-square distribution with v degrees of freedom is denoted 𝛸𝛸 𝑣𝑣 2 . The mean and variance of this distribution are equal to the number of degrees of freedom and twice the number of degrees of freedom E [ 𝛸𝛸 𝑣𝑣 2 ] = 𝑣𝑣 and Var [ 𝛸𝛸 𝑣𝑣 2 ] = 2 𝑣𝑣

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

View Full Document
This is the end of the preview. Sign up to access the rest of the document.

{[ snackBarMessage ]}

What students are saying

• 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.

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

• 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.

Dana University of Pennsylvania ‘17, Course Hero Intern

• 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.

Jill Tulane University ‘16, Course Hero Intern
{"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0}