Testing of Hypothesis

Testing of Hypothesis, Hypothesis testing, Independent t test, Independent z test, Analysis of variance, ANOVA, Comparison tests

Null and Alternative Hypothesis (2012)

by

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.

Null and Alternative Hypothesis

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

  1. 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 $H_1:\mu_1 < \mu_2$
  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
  3. 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 $H_1: \rho < 0$
  4. 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.

    $H_0: \beta_1 = 0$

    $H_1: \beta_1 \ne 0$
  5. 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
https://itfeature.com statistics help

MCQs about Number System (Intermediate Mathematics) Part-I

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

by

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!
Type I and Type II Errors

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

Visit Online MCQs Quiz Website

Statistics MCQs https://itfeature.com

Significance level in Statistics: Why do researchers use 5%?

by

Significance Level

The significance level in statistics is the level of probability at which it is agreed that the null hypothesis will be rejected. In academic research, usually, a 0.05 level of significance (level of significance) is used. The level of significance is also called a level of risk. Typical values for the selection of significance level range from 0.01 to 0.05, but other values can also be used depending on the context and the desired level of risk. A lower significance level means a lower probability of rejecting a true null hypothesis.

Significance Level in Statistics

The level of significance of an event (such as a statistical test) is the probability that the event will occur by chance. If the level is quite low then the probability of occurring that event by chance will be quite small. One can say that the event is significant as its occurrence is very small.

The significance level is the probability of rejecting the null hypothesis when it is true. In other words, the significance level is the probability of making a Type-I error, which is the error of incorrectly rejecting a true null hypothesis.

Significance Level in statistics

Type I Error

It has become part of the statistical hypothesis-testing culture.

  • It is a longstanding convention.
  • It reflects a concern over making type I errors (i.e., wanting to avoid the situation where you reject the null when it is true, that is, wanting to avoid “false positive” errors).
  • If you set the level of significance at 0.05, then you will only reject a true null hypothesis 5% of the time (i.e., you will only make a type-I error 5% of the time) in the long run.

The trade-off between Type-I and type-II Error

The choice of significance level is a trade-off between Type-I and Type-II errors. A smaller/ lower level of significance reduces the likelihood (probability) of Type-I errors (false positives) but increases the likelihood of Type-II errors (false negatives). In other words, the chance of type-I error increases for a higher significance level but decreases the chance of type-II error.

### Factors Affecting Significance Level:

  • Type of test: Different statistical tests have different formulas for calculating p-values, which are used to determine significance.
  • Sample size: Larger sample sizes generally lead to more powerful tests and lower p-values, making it easier to reject the null hypothesis.
  • Effect size: The magnitude of the difference between the null and alternative hypotheses can also influence the p-value.

In conclusion, the level of significance is a powerful tool that helps us to navigate the uncertainties in data analysis. By understanding the role of the significance level, one can make more wise informed decisions about the validity of research findings. In summary, the significance level is a crucial stage in the hypothesis testing procedure that helps the researchers make decisions about whether to accept or reject a null hypothesis based on the observed data. By carefully considering the significance level, researchers can balance the risk of making a Type-I error with the power of their tests.

Statistics Help: https://itfeature.com

Learn R Programming Language