Md. Siddique Hossain
Answer to the question no 01
Inference Regarding the population variance, σ 2
An important area of statistic is concern with making inference about the population variance. Knowledge of population variability is an important element of statistical analysis. Two possibilities arise For example.
A) For a car rental agency .
* Tires with low variability’s is preferred compared with durable lives with high variability. B) A bank policy favor a single waiting line that feeds into several tellers. * may remain same whether more than one lines formal.
* σ 2 may be lower in single line.
* σ 2 is higher in more than one line.
Like the population mean µ , σ 2 is ordinarily unknown, and its value must estimated using sample data.
A random sample of n observations drawn from a population with unknown mean and unknown varience σ 2 .Denote the sample x 1, x2 ,…….xn The population variance is the expectation
σ 2 = E [ ( x - µ) ] , Which sagely that we consider the mean of ( xi – ) and n observation Since µ is unknown the sample mean is used to compute a sample variance. The quantity
S2= 1/n-1i=1n (xi-) is called a sample variance ,and its square root s is called the sample standard deviation .
Given a specific random sample variance and the sample variance would be different for each random sample. P ( x2n-1 , 1-α/2 ≤ x2n-1 ≤ x2n-1 ,α/2)
n= 25 , n-1=24
P (x2n-1 < 12.40) =0.025 ,P (x2n-1 > 12.40)= 0.975,P(x2n-1 > 39.36) = 0.025,P (x2n-1 < 39.36) = 0.975
Chapter 09 Hypothesis
For a large sample size the sampling distribution of chi square can be closely approximated by a continuous curve known as the chi-squared distribution. If we can assume that a population distribution is normal, then it can be shown that the sample variance and the population variance are related through a probability distribution which is known as the chi-squared distribution.
Multiplying s² by (n-1) and dividing it by Ϭ² converts it into chi-squared random variable. It is possible to find the probabilities of this variable: α = p [ (n-1)s²/Ϭ² when the number of degrees of freedom is n – 1. Important properties of chi-squared distribution:
1. χ² distribution is a continuous probability which has the value zero at its lower limit and extends to infinity in the positive direction. 2. The exact shape of the distribution depends upon the number of degrees of freedom. When this value is small, the shape of the curve is skewed to the right and the distribution becomes more and more symmetrical as the value increases and thus can be approximated by the normal distribution. 3. The mean of the chi-squared distribution is given by, E(x²) = v and variance is given by, v(x²) = 2v.The chi square distribution with (n-1) degrees of freedom is the distribution of the squares of (n-1) independent standard normal variables. 4. As v gets larger, χ² approaches the normal distribution with mean v and standard deviation √2v. 5. The sum of independent χ² variables is also a χ² variable. Chi-square distribution of sample and population variances
Given a random sample of n observations from a normally distributed population whose population variance is Ϭ² and whose resulting sample variance is s². it can be shown that (n-1)s² ÷ Ϭ² = ∑(Xi – x)² ÷ Ϭ² has a distribution known as the chi-square distribution with n-1 degrees of freedom. We can use the properties of the chi-square distribution to find the variance of the sampling distribution of the sample variance when the parent population is normal.
Confidence interval for variance:
Confidence interval for variance σ is based on the sampling distribution of n-1s^2σ2 which follows χ^2 distribution with (n-1) degrees of freedom, 100(1-α)% confidence...