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STANDARD DEVIATIONS AND VARIANCES ESTIMATIONS

Estimation of ( and (2

Although there are a number of methods of estimating the standard deviation of a population the sample standard deviation is the most widely used estimator of this parameter. If we use s to make inferences about ( (or s2 to make inference about ( 2 ) the theory on which a confidence interval for ( is based requires that the population sampled has roughly the shape of a normal distribution, in which case the statistic called (2 has as its sampling distribution an important continuous distribution called the chi square distribution [pic] . In the formula as with the t distribution,(n-1) is called degrees of freedom. (2 Distribution has the following properties

1. It involves squared observations and hence it is always positive. Its value is always greater than or equal to zero. 2. The distribution is not symmetrical. It is skewed to the right so that its skewness is positive. However, as the number of degrees of freedom increases (2 approaches a symmetric distribution. In other words As the degrees of freedom increase, the chi-square curve approaches a normal distribution 3. Similar to t distribution there is a family of (2 distributions. There is a particular distribution for each degree of freedom. (2 is also called goodness- of fit test. Also used for analyzing qualitative variables such as opinions of persons, religious affiliations, and smoking habits etc. (2 test deals with judgements about proportions of two or more than two populations. The mean of the distribution is equal to the number of degrees of freedom: μ = v The variance is equal to two times the number of degrees of freedom: σ2 = 2 * v When the degrees of freedom are greater than or equal to 2, the maximum value for Y occurs when Χ2 = v – 2. The chi-square distribution is constructed so that the total area under the curve is equal to 1. The area under the curve between 0 and a particular chi-square value is a cumulative probability associated with that chi-square value. For example, in the figure below, the shaded area represents a cumulative probability associated with a chi-square statistic equal to A; that is, it is the probability that the value of a chi-square statistic will fall between 0 and A.

[pic]

In the figure below, the red curve shows the distribution of chi-square values computed from all possible samples of size 3, where degrees of freedom is n - 1 = 3 - 1 = 2. Similarly, the green curve shows the distribution for samples of size 5 (degrees of freedom equal to 4); and the blue curve, for samples of size 11 (degrees of freedom equal to 10). [pic]

The mean of a Chi Square distribution is its degrees of freedom. Chi Square distributions are positively skewed, with the degree of skew decreasing with increasing degrees of freedom. As the degrees of freedom increase, the Chi Square Distribution approaches a normal distribution. The figure above shows density functions for three Chi square distributions. Notice how the skew decreases as the degrees of freedom increases.

(

0 (2( (2

Unlike the normal and t distribution, the domain of the (2 distribution is restricted to the non-negative real numbers. [pic] is the value for which the area to the right under the [pic] is equal to(. Thus [pic] is such that the area to its right under the curve is (/2 , while (2 1-(/2 is such that the area to its left under the curve is (/2. We make the distinction because the (2 distribution is not symmetrical. Referring to the diagram below we can assert with probability 1- ( that a random variable having the (2 distribution will take on a value between (2 1-(/2 and (2 (/2 . Applying this result to the (2 statistic given above we can assert with...