What is statistical inference?
Statistical inference can be defined as the method of drawing conclusions from data which are subject to random variations. This is based on the mathematical laws of probability. Probability is the branch of statistics, where we make inferences from a finite set of observations to an infinite set of new observations. The finite set of observations is called as Samples and the infinite set is called as populations. Suppose let us consider tossing a few coins. Here the total number of outcomes i.e., getting the heads and tails are called as the sample space. The new observation that is getting particularly heads or tails is the population.
Solutions to statistical inference:
There are two kinds of statistical inferences. One is the estimation, where we use the sample and sample variables to predict the population variables. The second is the hypothesis testing. Here we use the samples and sample variables to test the population and the population variables.
Estimation:
Estimation can be divided into two types. One is point estimation and the other is interval estimation. In the point estimation we use the sample variable to estimate the parameter ?, whereas in the interval estimation we use the samples variable to construct an interval which is equated to p where p is the confidence level adopted.
The most common method adopted for point estimation is the maximum likelihood estimation (MLE) which consists of choosing the estimate that maximizes the probability of the statistical material. MLE is the best solution if the statistical material is large. Special cases of MLE are the sample mean dented as E(X) and the relative frequency denoted by P(X=x).
Illustrations of MLE:
A game is played with a single fair die. A player wins Rs.20 if a 2 turns up and Rs.40 if a 4 turns up, and he losses if a 6 turns up. While he neither wins nor loses if any other face turns up. Find the expected sum of money he can win.
Let X be the random variable denoting the amount he can win. The possible values are 20,40,-30,0
P[X=20] = P(getting 2) = 1/6
P[x=40] = P( getting 4 = )1/6
P[X = -30] = P(getting 6] = 1/6
The remaining probability is ½.
Hence the mean is E(X) = 20(1/6) + 40(1/6) + (-30)(1/6) +0(1/2) = 5.
Hence the expected sum of money he can win is Rs.5
Hypothetical Testing:
A statistical hypothesis is a statement of the numerical value of the population parameter.
The steps involved in solving a statistical hypothesis is
1. State the null hypothesis H_{o}
2. State the alternative hypothesis H_{a}
3. Specify the level of significance α
4. Determine the critical regions and the appropriate test statistic.
5. Compute the equivalent test statistic of the observed value of the parameter.
6. Take the decision either to reject H_{o} or accept H_{o.}