Testing of Hypothesis - Statistics for Data Analyst

Testing of Hypothesis or Hypothesis Testing

Jan 12, 2025Feb 22, 2012 by Muhammad Imdad Ullah

To whom is the researcher similar in hypothesis testing: the defense attorney or the prosecuting attorney? Why?

The researcher is similar to the prosecuting attorney in the sense that the researcher brings the null hypothesis “to trial” when she believes there is a probability of strong evidence against the null.

Just as the prosecutor usually believes that the person on trial is not innocent, the researcher usually believes that the null hypothesis is not true.
In the court system, the jury must assume (by law) that the person is innocent until the evidence calls this assumption into question; analogously, in hypothesis testing the researcher must assume (to use hypothesis testing) that the null hypothesis is true until the evidence calls this assumption into question.

The world aournd us is complex enough and full of uncertainty. Onlyobserving the data can not tell us if a pattern or relationship exists, or if it is just due to random chance. Therefore, we need hypthesis testing procedure that provides us a systematic method to analyze the sample data and draw conclusions (or make wise decisions) about a larger population, with a clear understanding of the likelihood of being wrong.

In conclusion, like statistical estimation, the statistical hypothesis testing is a cornerstone of statistical analysis. It provides a way to move beyond simply observing data and allows us to draw meaningful inferences about populations, evaluate claims, and make informed decisions in the face of uncertainty.

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Null and Alternative Hypothesis (2012)

Sep 11, 2024Feb 13, 2012 by Muhammad Imdad Ullah

Specifying the Null and Alternative Hypothesis of the following Statistical Tests:

1) The t-test for independent samples,
2) One-way analysis of variance,
3) The t-test for correlation coefficients?
4) The t-test for a regression coefficient.
5) Chi-Square Goodness of Fit Test

Before writing the Null and Alternative Hypothesis for each of the above, understand the following in general about the Null and Alternative hypothesis.
In each of these, the null hypothesis says there is no relationship or no difference. The alternative hypothesis says that there is a relationship or there is a difference. The null hypothesis of a test always represents “no effect” or “no relationship” between variables, while the alternative hypothesis states the research prediction of an effect or relationship.

The Null and Alternative hypothesis for each of the above is as follows:

In this case, the null hypothesis says that the two population means (i.e., $\mu_1$ and $\mu_2$) are equal; the alternative hypothesis says that they are not equal.

$H_0: \mu_1 = \mu_2$

$H_1: \mu_1 \ne \mu_2$ or $H_1:\mu_1 > \mu_2$ or $H_1:\mu_1 < \mu_2$
In this case, the null hypothesis says that all of the population means are equal; the alternative hypothesis says that at least two of the means are not equal. If there are 4 populations to be compared then

$H_0: \mu_1=\mu_2=\mu_3 = \mu_4$

$H_1:$ at least two population means are different
In this case, the null hypothesis says that the population correlation (i.e., $\rho$) is zero; the alternative hypothesis says that it is not equal to zero.

$H_0: \rho = 0$

$H_1: \rho \ne 0$ or $H_1: \rho > 0$ or $H_1: \rho < 0$
In this case, the null hypothesis says that the population regression coefficient ($\beta$) is zero, and the alternative says that it is not equal to zero.

$H_0: \beta_1 = 0$

$H_1: \beta_1 \ne 0$
In this case, the null hypothesis says that there is no association between categories of Variable-1 and categories of variable-2. The alternative hypothesis says that there is an association between categories of Variable-1 and categories of Variable-2.

$H_0:$ There is no association between grouping variables

$H_1:$ There is an association between grouping variables

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Type I and Type II Errors in Statistics: A Quick Guide

Sep 18, 2024Feb 12, 2012 by Muhammad Imdad Ullah

In hypothesis testing, two types of errors can be made: Type I and Type II errors.

Type I and Type II Errors

A Type I error occurs when you reject a true null hypothesis (remember that when the null hypothesis is true you hope to retain it). Type-I error is a false positive error.
α=P(type I error)=P(Rejecting the null hypothesis when it is true)
Type I error is more serious than type II error and therefore more important to avoid than a type II error.
A Type II error occurs when you fail to reject a false null hypothesis (remember that when the null hypothesis is false you hope to reject it). Type II error is a false negative error.
$\beta$=P(type II error) = P(accepting null hypothesis when alternative hypothesis is true)
The best way to allow yourself to set a low alpha level (i.e., to have a small chance of making a Type I error) and to have a good chance of rejecting the null when it is false (i.e., to have a small chance of making a Type II error) is to increase the sample size.
The key to hypothesis testing is to use a large sample in your research study rather than a small sample!

If you do reject your null hypothesis, then it is also essential that you determine whether the size of the relationship is practically significant.
Therefore, the hypothesis test procedure is adjusted so that there is a guaranteed “low” probability of rejecting the null hypothesis wrongly; this probability is never zero.

Therefore, for type I and Type II errors remember that falsely rejecting the null hypothesis results in an error called Type-I error and falsely accepting the null hypothesis results in Type-II Error.

Read more about Level of significance in Statistics

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