Confidence Interval for the Variance of a Normal Population Bernd - PowerPoint PPT Presentation

χ 2 -Distribution Confidence Interval for the Variance Confidence Interval for the Variance of a Normal Population Bernd Schr¨ oder logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science Confidence Interval for the Variance of a Normal Population

χ 2 -Distribution Confidence Interval for the Variance Theorem. logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science Confidence Interval for the Variance of a Normal Population

χ 2 -Distribution Confidence Interval for the Variance Theorem. The distribution of the sample variances S 2 of samples from a normal distribution. logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science Confidence Interval for the Variance of a Normal Population

χ 2 -Distribution Confidence Interval for the Variance Theorem. The distribution of the sample variances S 2 of samples from a normal distribution. If n values are sampled from a normal distribution with standard deviation σ logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science Confidence Interval for the Variance of a Normal Population

χ 2 -Distribution Confidence Interval for the Variance Theorem. The distribution of the sample variances S 2 of samples from a normal distribution. If n values are sampled from a normal distribution with standard deviation σ , then the random variable logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science Confidence Interval for the Variance of a Normal Population

χ 2 -Distribution Confidence Interval for the Variance Theorem. The distribution of the sample variances S 2 of samples from a normal distribution. If n values are sampled from a normal distribution with standard deviation σ , then the random variable ( n − 1 ) S 2 σ 2 logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science Confidence Interval for the Variance of a Normal Population

χ 2 -Distribution Confidence Interval for the Variance Theorem. The distribution of the sample variances S 2 of samples from a normal distribution. If n values are sampled from a normal distribution with standard deviation σ , then the random variable ( n − 1 ) S 2 σ 2 has a chi-squared distribution ( χ 2 -distribution) with n − 1 degrees of freedom. logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science Confidence Interval for the Variance of a Normal Population

χ 2 -Distribution Confidence Interval for the Variance Theorem. The distribution of the sample variances S 2 of samples from a normal distribution. If n values are sampled from a normal distribution with standard deviation σ , then the random variable ( n − 1 ) S 2 σ 2 has a chi-squared distribution ( χ 2 -distribution) with n − 1 degrees of freedom. Remember that the χ 2 -distribution is a Gamma distribution with α = n 2 and β = 2. logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science Confidence Interval for the Variance of a Normal Population

χ 2 -Distribution Confidence Interval for the Variance Definition. logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science Confidence Interval for the Variance of a Normal Population

χ 2 -Distribution Confidence Interval for the Variance Definition. The function n 2 − 1 e − x � 1 2 x 2 ; for x > 0 , n Γ ( n 2 ) 2 f n ( x ) : = 0; otherwise , is the density of the χ 2 -distribution with n degrees of freedom . logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science Confidence Interval for the Variance of a Normal Population

χ 2 -Distribution Confidence Interval for the Variance Definition. The function 2 − 1 e − x n � 1 2 x 2 ; for x > 0 , n Γ ( n 2 ) 2 f n ( x ) : = 0; otherwise , is the density of the χ 2 -distribution with n degrees of freedom . Theorem. logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science Confidence Interval for the Variance of a Normal Population

χ 2 -Distribution Confidence Interval for the Variance Definition. The function 2 − 1 e − x n � 1 2 x 2 ; for x > 0 , n Γ ( n 2 ) 2 f n ( x ) : = 0; otherwise , is the density of the χ 2 -distribution with n degrees of freedom . Theorem. Let C n be a random variable with a χ 2 -distribution with n degrees of freedom. logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science Confidence Interval for the Variance of a Normal Population

χ 2 -Distribution Confidence Interval for the Variance Definition. The function n 2 − 1 e − x � 1 2 x 2 ; for x > 0 , n Γ ( n 2 ) 2 f n ( x ) : = 0; otherwise , is the density of the χ 2 -distribution with n degrees of freedom . Theorem. Let C n be a random variable with a χ 2 -distribution with n degrees of freedom. Then E ( C n ) = n logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science Confidence Interval for the Variance of a Normal Population

χ 2 -Distribution Confidence Interval for the Variance Definition. The function n 2 − 1 e − x � 1 2 x 2 ; for x > 0 , n Γ ( n 2 ) 2 f n ( x ) : = 0; otherwise , is the density of the χ 2 -distribution with n degrees of freedom . Theorem. Let C n be a random variable with a χ 2 -distribution with n degrees of freedom. Then E ( C n ) = n V ( C n ) = 2 n . and logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science Confidence Interval for the Variance of a Normal Population

χ 2 -Distribution Confidence Interval for the Variance Definition. logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science Confidence Interval for the Variance of a Normal Population

χ 2 -Distribution Confidence Interval for the Variance Definition. For 0 < α < 1 , the number χ 2 α ( n ) is the unique real number such that the area that is to the right of χ 2 α ( n ) under the density of the χ 2 -distribution with n degrees of freedom is α . logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science Confidence Interval for the Variance of a Normal Population

χ 2 -Distribution Confidence Interval for the Variance Definition. For 0 < α < 1 , the number χ 2 α ( n ) is the unique real number such that the area that is to the right of χ 2 α ( n ) under the density of the χ 2 -distribution with n degrees of freedom is α . ✲ logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science Confidence Interval for the Variance of a Normal Population

χ 2 -Distribution Confidence Interval for the Variance Definition. For 0 < α < 1 , the number χ 2 α ( n ) is the unique real number such that the area that is to the right of χ 2 α ( n ) under the density of the χ 2 -distribution with n degrees of freedom is α . ✻ ✲ logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science Confidence Interval for the Variance of a Normal Population

χ 2 -Distribution Confidence Interval for the Variance Definition. For 0 < α < 1 , the number χ 2 α ( n ) is the unique real number such that the area that is to the right of χ 2 α ( n ) under the density of the χ 2 -distribution with n degrees of freedom is α . ✻ ✲ logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science Confidence Interval for the Variance of a Normal Population

χ 2 -Distribution Confidence Interval for the Variance Definition. For 0 < α < 1 , the number χ 2 α ( n ) is the unique real number such that the area that is to the right of χ 2 α ( n ) under the density of the χ 2 -distribution with n degrees of freedom is α . ✻ ✲ logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science Confidence Interval for the Variance of a Normal Population

χ 2 -Distribution Confidence Interval for the Variance Definition. For 0 < α < 1 , the number χ 2 α ( n ) is the unique real number such that the area that is to the right of χ 2 α ( n ) under the density of the χ 2 -distribution with n degrees of freedom is α . ✻ α 2 ✲ logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science Confidence Interval for the Variance of a Normal Population

χ 2 -Distribution Confidence Interval for the Variance Definition. For 0 < α < 1 , the number χ 2 α ( n ) is the unique real number such that the area that is to the right of χ 2 α ( n ) under the density of the χ 2 -distribution with n degrees of freedom is α . ✻ α 2 ✲ χ 2 2 (n-l) α logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science Confidence Interval for the Variance of a Normal Population

χ 2 -Distribution Confidence Interval for the Variance Definition. For 0 < α < 1 , the number χ 2 α ( n ) is the unique real number such that the area that is to the right of χ 2 α ( n ) under the density of the χ 2 -distribution with n degrees of freedom is α . ✻ α 2 ✲ χ 2 2 (n-l) α logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science Confidence Interval for the Variance of a Normal Population