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1. ESTIMATION: (point and interval): Definition: Estimation is an important aspect of statistical inference which deals with obtaining an estimate of a given parameter of a distribution either by computing a single estimate or an interval from a point. The parameter we tend to estimate is called the ESTIMAND. Estimation is of two types. 1.Point Estimation 2.Interval Estimation. 1. Point Estimation: It is gotten from observed values of a random sample to obtain a single number close to the unknown parameter. 2.Interval Estimation: It is an estimation of a population parameter given by two numbers b/w which the parameters may be considered to lie. Interval estimation indicates precision or accuracy of an estimate and are therefore preferable to point estimates. Properties of Estimation: 1.Unbiasedness: this is when the expected value is equal to the parameter estimated. Example:1 Show that the mean ?̅ is an unbiased estimator of ?. Solution in Video. 2.Consistency: if the estimated values comes closer to the desired output as sample sizes increase. 3.Efficiency. 4.Mean square Error. Interval Estimation. Here as said b/4 our answer is an interval which contains the desired output. The end points of the intervals (?,?)is called the confidence limits. Upper Confident limit We have several cases of estimation. Which include; 1.Estimation of mean (with known variance) The formula is used to solve this type of questions The example below is solved in the video Lower Confident limit

2. 2.Estimation of mean (large number n>=30) The only difference here is that we replace ?2 with ?2 and basically do the same thing. Solution in video. 3.Estimation of mean (small sample) Here we make use of ? instead of ?. So basically, whenever you have a small sample size, then we will have a degree of freedom here ? − 1. Example Solution in video. 4.Confidence interval for difference of two mean. Given two mean and variance, we will make use of the following formula Solution in video. Example A random sample Standard deviation Taken from a Mean ?̅=75

3. Topic II Test of Significance/Hypothesis: Hypothesis is a claim about a population that can be subjected to investigation. The null Hypothesis is the one being tested. While the alternative hypothesis contradicts the null hypothesis. Decision error We have two types of error Type I and Type II errors respectively. The probability of Type I error is ? while that of type II is ?. Accept ?0 Correct decision TYPE II error Reject ?0 TYPE I error Correct decision ?0 is True ?0 is False Types of test. The null hypothesis always goes with equality while the alternative hypothesis is the other way round Example: I. II. These contradicts the null hypothesis in one direction. Hence, we call it a one-tailed-test to the right or left, depending on the direction of the inequality ?0:? > ?0 ?0:? < ?0 ?0:? ≠ ?0 Whenever we have something like this it is call a two-way-test. III. Problems on Hypothesis: 1.Test concerning one population proportion. (for ? = ??) The test Statistics is

4. Decision rule: ?0 is rejected if |?| > ?? 2 for ?0:? = ?0 against ?1:? > ?0 the critical region is to the right. ?0 is rejected if ? > ?? and in testing against ?0:? < ?0, the critical region is to the left and null hypothesis is rejected if ? < −?? 2.Test for the difference B/W two proportions: The test statistic for the hypothesis is given as follows. 3.Test concerning one population mean. The test statistic is given as follows