1 Estimation of Population Mean Dr. T. T. Kachwala

Estimation of Population Characteristics from Sample Characteristics 2 Statistical inference is based on estimation and hypothesis testing. In both, we shall be making inferences about the characteristics of populations from information contained in samples. To calculate exact population mean would be an impossible goal. However, we can make an estimate with a statement about the error that will accompany this estimate. Any sample statistic that is used to estimate a population parameter is called an estimator i.e. an estimator is a sample statistic used to estimate a population parameter.

Estimator & its desirable properties 3 For example sample mean ' ' is an estimator of the population mean . A good estimator is unbiased and efficient. An unbiased estimator is an estimator such that mean of its sampling distribution = population mean. For example A.M is unbiased because. On the other hand efficiency refers to the size of standard error of the statistic. Amongst the unbiased estimators, the estimator with the least variance is an efficient estimator.

Unbiased & Efficient Estimators 4 On similar lines, Standard Deviation s is unbiased and efficient estimator of  Using '(n-1)' in the denominator for ‘s’ gives us an unbiased estimator of  { i.e. if we used the formula; if can be proved that the estimator 's' will be a biased estimator of the population standard deviation  }

Point Estimate & Interval Estimate 5 There are two types of estimates about a population 1.Point estimate 2.Interval estimate A point estimate is a single number that is used to estimate an unknown population parameter. It is often insufficient because it is either right or wrong for e.g. the attendance in class of 60 students today will be 52 students. Point estimate is accompanied by an estimate of the error that might be involved. For example 52  5 On the other hand, an interval estimate is a range of values used to estimate a population parameter. For e.g. " ”. It indicates the error in two ways: by the extent of its range and by the probability of the true population parameter lying within that range.

Interval Estimates-Basic Concepts 6 An interval estimate describes a range of values within which a population parameters is likely to lie One can obtain intervals of different widths using the following mathematical formula: Interval estimate = Point estimate ± Margin of error Where 'z' is the table value from Standard Normal Table. For different values of z one can obtain different interval width {the value of z lies between 0.01 to 3.09 to infinity}. Also for each value of 'z' one can obtain the corresponding value of 'a' using Standard Normal Table.

Interval Estimates-Basic Concepts 7 There is a direct relationship existing between confidence level and the width of interval estimate - for example: Value of z Width of Interval Estimate Confidence level z = % Confidence level z = % Confidence level z = % Confidence level In statistics, the probability that we associate with an interval estimate is called the confidence level. It indicates how confident we are that the interval estimate will include the population parameter. The most frequently used confidence intervals are 90 percent, 95 percent and 99 percent confidence intervals. The corresponding three popular values of z are: 1.645, 1.96 &

Interval Estimates using the 't' Distribution 8 When the population standard derivation '  ' is not known and the sample size is '30 or less’; then normal distribution is not the appropriate sampling distribution. Fortunately, another distribution exists that is appropriate in these cases. It is called the 't' distribution. Statistician 'Gosset' who adopted the pen name 'student' developed the t distribution commonly referred as student's t distribution. Use of t distribution for estimating is required when: the sample size is 30 or less Population standard deviation is not known Population is normal or approximately normal The ‘t’ distribution like the normal distribution is symmetrical. There is a different t distribution for each of the possible “level of significance” and "degree of freedom".

9 Concept of Level of Significance α Mathematically Level of Significance α = 1 – Confidence level For example 95% confidence level is the same as 5% level of significance Level of Significance α signifies the total area under the tail of the distribution

Concept of Degree of Freedom 10 Degree of Freedom can be defined as the number of values we can choose freely. Alternatively, by degree of freedom we mean the number of independent observations that can be assigned arbitrarily without violating the restrictions or limitations placed. Symbolically, = N - K (nu) (observn) (constraint) In general, for problems concerning one sample of size 'n' = n - 1 and for problems concerning two samples of size n 1 & n 2, ν = n 1 + n 2 – 2

Characteristics of the ‘t’ distribution 11 The 't' table differs from 'z' table. Because there is a different t distribution for each numbers of degree of freedom, t values are given only for a few percentage (level of significance for e.g. :  = 0.1, 0.05, 0.01) Another difference in the 't' table is that it focuses on rejection area (level of significance  = 0.05, 0.01) while z table focuses on acceptance area (confidence level = 95%, 99%) Finally, while specifying t table, we must specify level of significance and degree of freedom for e.g. t 0.05, 10 = 2.22

Relationship between 't' distribution and Normal distribution 12 Both 't' distribution and Normal distribution are symmetrical. In general, 't' distribution is flatter than the normal distribution and there is a different 't' distribution for every degree of freedom. As sample size gets larger t distribution approximates normal distribution as indicated below: (t distribution n>30) Normal Distribution t distribution n = 2

Relationship between 't' distribution and Normal distribution 13 A t distribution is lower at the mean and higher at the tails than a normal distribution. A t distribution has proportionately more of its area in its tail than the normal distribution. As a result, the ordinates have to go further out from the mean of a t distribution to include the same area under the curve. Interval widths from t distribution are wider than those based on normal distribution.

14 Thank You Dr. T. T. Kachwala