What is null hypothesis

1. Sampling distribution:

Statistic is a random variable and its probability distribution is called sampling distribution.Parameter is a quantitative characteristic, which indexes/identifies the respective distribution.

2. Random sample:

Any set of realisations $X_1, X_2 ,\dots , X_n$ made on $X$ under independent and identical conditions is called a random sample.The statistic itself is a random variable and has a probability distribution.

3. Standard error: 

$SE\left(X\right)=\frac{\sigma }{\sqrt{n}}$

4. Hypothesis:

(i) Null hypothesis: 

It is a hypothesis which is tested for possible rejection. Statistical test leads to take decision on the null hypothesis.

(ii) Alternative hypothesis:

A statement about the population, which contradicts the null hypothesis, depending upon the situation, is called alternative hypothesis.

5. Rejection rule:

The decision rule leading to rejection of $H_0$  is called as rejection rule.

6. Types of errors:

(i) Type I error is rejecting the true null hypothesis.

(ii) Type II error is not rejecting a false null hypothesis.

7. Level of significance:

Upper limit of the $\mathrm{P}\text{(Type/error)}$ is called level of significance, denoted by $\alpha$.

8. Critical region:

Critical region is a subset of the sample space defined by the rejection rule.Critical value identifies the elements of critical region.

9. One-tailed test:

(i) A Hypothesis Test where the rejection region is located to the extreme right of the distribution is called right-tailed test.

(ii) A Hypothesis Test where the rejection region is located to the extreme left of the distribution is called left-tailed test.

10. Two-tailed test:

A test of a statistical hypothesis, where the region of rejection is on both sides of the sampling distribution, is called a two-tailed test.

11. General procedure for test of hypothesis:

(i) Framing hypothesis.

(ii) Describe the sample i.e., data.

(iii) Specify he desired level of significance $\alpha$.

(iv) Specify the test statistic and its sampling distribution under $H_0$.

(v) Calculate the value of the test statistic under $H_0$.

(vi) Find the critical value(s) from the statistical table generated from the sampling distribution of the test statistic under $H_0$ corresponding to $\alpha$.

(viii) Decide on rejecting or not rejecting the null hypothesis based on the rejection rule which compares the calculated value(s) with the table value(s).

12. Large samples:

For hypotheses testing based on two samples, if the sizes of both the samples are greater than or equal to $30$, they are called large samples.

13. Test of hypotheses:

(i) For population mean test statistics is $Z=\frac{\overline{X}-\mu }{{\displaystyle \frac{\sigma }{\sqrt{n}}}}$ under $H_0$

(ii) For population mean when population variance is unknown, test statistic is $Z=\frac{\overline{X}-\mu }{{\displaystyle \frac{s}{\sqrt{n}}}}$ under $H_0$

(iii) For equality of means of two populations when population variances are known, test statistics is $Z=\frac{\left(\overline{X}-\overline{Y}\right)}{\sqrt{\frac{{{\sigma }^2}_x}{m}+\frac{{{\sigma }^2}_{\gamma }}{n}}}$ under $H_0$.

(iv) For equality of means of two populations when population variances are unknown, test statistics is $Z=\frac{(\overline{X}-\overline{Y})}{\sqrt{\frac{{s^2}_x}{m}+\frac{{s^2}_y}{n}}}$ under $H_0$.

(v) For population proportions, test statistic is $Z=\frac{\overline{p}-P}{\sqrt{\frac{PQ}{n}}}$ under $H_0$.

(vi) For equality of proportions of two population is $Z=\frac{P_x-P_{\gamma }}{\sqrt{pq\left(\frac{1}{m}+\frac{1}{n}\right)}}$ under $H_0$.