Testing of Hypothesis

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

P-value Interpretation and Misinterpretation of P-value 2012

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The P-value is a probability, with a value ranging from zero to one. It is a measure of how much evidence we have against the null hypothesis. P-value is a way to express the likelihood that $H_0$ is not true. The smaller the p-value, the more evidence we have against $H_0$. Here we will discuss about P-value and Its Interpretation.

P-value Definition

The largest significance level at which we would accept the null hypothesis. It enables us to test the hypothesis without first specifying a value for $\alpha$. OR

The probability of observing a sample value as extreme as, or more extreme than, the value observed, given that the null hypothesis is true.

p value and significance level

P-value Interpretation

In general, the P-value interpretation is “If the P-value is smaller than the chosen significance level then $H_0$ (null hypothesis) is rejected even when it is true. If the P-value is larger than the significance level $H_0$ is not rejected”.

p-value Interpretation

If the P-value is less than

  • 0.10, we have some evidence that $H_0$ is not true
  • 0.05, strong evidence that $H_0$ is not true
  • 0.01, Very strong evidence that $H_0$ is not true
  • 0.001, extremely strong evidence that $H_0$ is not true

Misinterpretation of a P-value

Many people misunderstand P-values. For example, if the P-value is 0.03 then it means that there is a 3% chance of observing a difference as large as you observed even if the two population means are the same (i.e. the null hypothesis is true). It is tempting to conclude, therefore, that there is a 97% chance that the difference you observed reflects a real difference between populations and a 3% chance that the difference is due to chance. However, this would be an incorrect conclusion. What you can say is that random sampling from identical populations would lead to a difference smaller than you observed in 97% of experiments and larger than you observed in 3% of experiments.

Note that p-values are a valuable tool in hypothesis testing, but they should be used thoughtfully and in conjunction with other analyses.

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P value and Significance Level

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Difference Between the P value and Significance Level?

Basically in hypothesis testing the goal is to see if the probability value is less than or equal to the significance level (i.e., is p ≤ alpha). It is also called the size of the test or the size of the critical region. It is generally specified before any samples are drawn so that the results obtained will not influence our choice.

p value and significance level

The difference between P Value and Significance Level is

  • The probability value (also called the p-value) is the probability of the observed result found in your research study occurring (or an even more extreme result occurring), under the assumption that the null hypothesis is true (i.e., if the null were true).
  • In hypothesis testing, the researcher assumes that the null hypothesis is true and then sees how often the observed finding would occur if this assumption were true (i.e., the researcher determines the p-value).
  • The significance level (also called the alpha level) is the cutoff value the researcher selects and then uses to decide when to reject the null hypothesis.
  • Most researchers select the significance or alpha level of 0.05 to use in their research; hence, they reject the null hypothesis when the p-value is less than or equal to 0.05.
  • The key idea of hypothesis testing is that you reject the null hypothesis when the p-value is less than or equal to the significance level of 0.05.
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Testing of Hypothesis or Hypothesis Testing Made Easy

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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.
Hypothesis Testing

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)

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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
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