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Estimate is a numerical value of the estimator computed from a given set of sample values.

The set of all those values of the test statistic, which lead to the rejection of the null hypothesis, is called critical region (ω). 

The Critical Region is also called as Rejection Region. 

The value of test statistic, which separates the critical region (ie. rejection region) and . acceptance regions is called the Critical value, denoted by + k or -k or – k₁ / + k_2.

‘While estimating the unknown parameter, if a specific’value is proposed as an estimate, which is called Point estimation’. 

While estimating the unknown parameter instead of a specific value, an interval is proposed, which is likely to contain the parameter is called Interval estimation'.

A Confidence interval is a interval within which the unknown population parameter is expected to lie.

It is a statement or an assertion made regarding the parameters.

Statistical constants of the population such as Mean (µ), S.D (σ) are called parameter. 

Ex:- The population has mean = µ, 

S.D = σ, population proportion P₀ 

Statistical measures computed from the samples such as Mean (x̄), S.D(σ) are called statistic. 

Ex:- A random sample of size ‘n’ has mean=x̄ and S.D = s, sample proportion p

(n – 1 – 2) = (n – 3)d.f.

Here Z_cal < Z_tab. (H₀) Null hypothesis is accepted.

(a) Observations should be independent,0.6 

(b) Total frequency N should be large, (e) Each Ei‘s ≥ 5.

• if the critical region is considered at one tail of the null distribution of the test statistic, the test is one tailed. 
• if the critical region is considered at both the tails of the null distribution of the test statistic, the test is two tailed.

Mean of (p₁ – p₂₎ = E(p₁ – p₂) 

= 0.3 – 0.2 = 0.1.

Statistics is a type of analysis that utilizes quantified methods and representations for a particular set of data. It uses different techniques in the gathering, analysis, and drawing of conclusions from a specific set of data in a numerical form. Statistics is often regarded as being a means by which observations are expressed numerically in order to investigate casual relationship between the variables. Any fact, to be called statistics must be numerically expressed (so that it can be counted, divided or be subject to mathematical analysis) and should be placed in relation to each other. But qualitative data cannot be included in statistics unless they are quantified by assigning some figures for assessment. However, not all numbers are comparable and measurable.
For example : The fact that height of a student is five feet tells nothing, unless it is comparable. Thus for figures to be included in statistics, they must be aggregate of facts and not individual figures. That why it is rightly said that statistics are figures but all figures are not statistics.

Given N=150, X = 78, n = 35

Population proportion 

= P₀ = X/N = \(\frac{78}{150}\) 

= 0.52; Q₀ = 1 -P₀= 0.48

Sample proportion = p = p₀ = 0.52

Standard error of sample proportion is

SE = \(\sqrt{n}\) = \(\sqrt\frac{0.52 \times 0.48}{35}\)

 = 0.084.

Probability of occurrence of type I Error is called level of significance, denoted by ‘α’. It is also called as Size of the test.

It is the probability of rejecting the null hypothesis when it is not true. (1 – β ), 

Here β = P (Type II Error) which is referred as ‘consumers risk.

Type I Error is taking a wrong decision to reject the null hypothesis when it is actually true. 

Type II Error is taking a wrong decision to accept the null hypothesis when it is actually not true.

1. It is used in interval estimation, to write down the confidence intervals. 

2. It is used in testing of hypothesis, to test whether the difference between the sample statistic and the population parameter is significant or not.

The standard deviation of the sampling distribution of sample statistic is called S.E of statistic.

A hypothesis, which completely specifies the parameter of the distribution, is a simple hypothesis. 

Ex:- H: µ = µ₀ (25) is a is a simple hypothesis, 

H: The population is normally distributed with mean µ = 25 and. σ = 2 

A hypothesis, which does not completely specify the parameter of the distribution, is a composite hypothesis. 

Ex:- H: The population is normally distributed with mean µ = 25

Null Hypothesis is a hypothesis, which is being tested for possible rejection under the assumption that it is true. Denoted by H₀ 

The hypothesis, which is accepted when the null hypothesis is rejected, is called alternative hypothesis. Denoted by H₁.

Given Z = 2.63 and Z_tab = 2.58. 

Here Z_cal > Z_tab. Null hypothesis is rejected.

The set of all samples of size ‘n’ that can be drawn from population is called sample space(S)

 The set of all the admissible values of the population parameters is called parameter space.

(a) Observations should be independent, 

(b) Total frequency N should be large, 

(c) Each cell frequencies ie ,a, b, c, d are all ≥ 5.

Here Z_cal < Z_tab H₀ is rejected and H₁ is accepted.

Each cell frequency ie. E, is ≥ 5.

The distribution of values of the statistic for different samples of the same size is called sampling distribution of a statistic.

Probability of occurrence of type I Error is called level of significance, denoted by ‘α’ 

The confidence interval is the interval within which the unknown population parameter is expected to lie. The prob. that the confidence interval contains the parameter is called confidence coefficient.