A the sample mean suppose in a simple random sample

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a. The sample Mean: Suppose in a simple random sample of size n picked up from a population, and then the sample mean represented by x is defined as: n x . . . . . x x x x n 3 2 1 The sample may be selected with replacement or without replacement. In the former, a number occurring more than once is accepted. A unit is repeated as many times as a random number occurs. In the latter, a random number is omitted at any subsequent stage. The sampling with replacement and without replacement is referred to infinite population and finite population respectively. The expection value of sample mean is the same as the population mean. Thus: E ( x ) = E ( x )= x/n E ( ) = x/N b. The Sample Variance: Suppose, the simple random sample of size n chosen from a population, the sample variance is used to estimate the population variance. In an equation form, rswor n 1 - N n - N S rswor N .] . . . . ) - (x [ rswr n )] x ( - x [ S rswr N ] ) - (x [ 2 2 2 2 2 2 2 Standard Error
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73 The standard deviation measures variability variable. The standard deviation of a sampling distribution is referred to standard error (S.E). It measures only sampling variability which occurs due to chance or random forces, in estimating a population parameter. The word error is used in place of deviation to emphasize that the variation among sample statistic is due to sampling errors. If is not known, we use the standard error given by: SE = S/ n Where S = n ) x - x ( 2 If n is large S = 1 n ) x - x ( 2 If n is small In drawing statistical inferences, the standard error is of great significance due to: 1) That it provides an idea about the reliability of sample. The lesser the standard error, the lesser the variation of population value from the expected (sample) value. Hence is grater reliability of sample. 2) That it helps to determine the confidence limits within which the parameter value is expected to lie. For large sample, sampling distribution tends to be close to normal distribution. In normal distribution, a range of mean one standard error, of mean two standard error, of mean three 3 standard error will give 68.27 per cent, 95.45 per cent and 99.73 per cent values respectively. The chance of a value lying outside 3 S.E is only 0.27 per cent i.e. approximately 3 in 1000. 3) That it aids in testing hypothesis and in interval estimation. 4) That it may be considered as the key to the sampling theory. Estimation Theory A technique which is used for generalizing the results of the sample to the population for estimating population parameters along with the degree of confidence is provided by an important branch of statistics is called statistical inference. In other words it is the process of inferring information about a population from a sample. This statistical inference deals with two main problems namely (a) estimation and (b) testing hypothesis.
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