Testing of Hypothesis

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

Type I and Type II Errors Examples

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The post covers the Type I and Type II Errors examples.

Hypothesis testing helps us to determine whether the results are statistically significant or occurred by chance. Hypothesis testing is based on probability, therefore, there is always a chance of making the wrong decision about the null hypothesis (a hypothesis about population). It means that there are two types of errors (Type I and Type II errors) that can be made when drawing a conclusion or decision.

Errors in Statistical Decision-Making

To understand the errors in statistical decision-making, we first need to see the step-by-step process of hypothesis testing:

  1. State the null hypothesis and the alternative hypothesis.
  2. Choose a level of significance (also called type-I error).
  3. Compute the required test statistics
  4. Find the critical value or p-value
  5. Reject or fail to reject the null hypothesis.

When you decide to reject or fail to reject the null hypothesis, there are four possible outcomes–two represent correct choices, and two represent errors. You can:
• Reject the null hypothesis when it is actually true (Type-I error)
• Reject the null hypothesis when it is actually false (Correct)
• Fail to reject the null hypothesis when it is actually true (Correct)
• Fail to reject the null hypothesis when it is actually false (Type-II error)

These four possibilities can be presented in the truth table.

Type I and Type II Errors Examples

Type I and Type II Errors Examples: Clinical Trial

To understand Type I and Type II errors, consider the example from clinical trials. In clinical trials, Hypothesis tests are often used to determine whether a new medicine leads to better outcomes in patients. Imagine you are a data professional and working in a pharmaceutical company. The company invents a new medicine to treat the common cold. The company tests a random sample of 200 people with cold symptoms. Without medicine, the typical person experiences cold symptoms for 7.5 days. The average recovery time for people who take the medicine is 6.2 days.

You conduct a hypothesis test to determine if the effect of the medicine on recovery time is statistically significant, or due to chance.

In this case:

  • Your null hypothesis ($H_0$) is that the medicine has no effect.
  • Your alternative hypothesis ($H_a$) is that the medicine is effective.

Type I Error

A Type-I error (also known as a false positive) occurs when a true null hypothesis is rejected. In other words, one can conclude that the result is statistically significant when in fact the results occurred by chance. To understand this, let in your clinical trial, the results indicate that the null hypothesis is true, which means that the medicine has no effect. In case, you make a Type-I error and reject the null hypothesis, it means that you incorrectly conclude that the medicine relieves cold symptoms while the medicine was (actually) ineffective.

The probability of making a Type I error is represented by $\alpha$ (the level of significance. Typically, a 0.05 (or 5%) significance level is used. A significance level of 5% means you are willing to accept a 5% chance you are wrong when you reject the null hypothesis.

Reduce the risk of Type I error

To reduce your chances of making Type I errors, it is advised to choose a lower significance level. For example, one can choose the significance level of 1% instead of the standard 5%. It will reduce the chances of making a Type I error from 5% to 1%.

Type II Error

A Type II error occurs when we fail to reject a null hypothesis when it is false. In other words, one may conclude that the result occurred by chance, however, in fact, it didn’t. For example, in a clinical study, if the null hypothesis is false, it means that the medicine is effective. In case you make a Type II Error and fail to reject the null hypothesis, it means that you incorrectly conclude that the medicine is ineffective while in reality, the medicine relieves cold symptoms.

The probability of making a Type II error is represented by $\beta$ and it is related to the power of a hypothesis test (power = $1- \beta$). Power refers to the likelihood that a test can correctly detect a real effect when there is one.

Note that reducing the risk of making a Type I error means that it is more likely to make a Type II error or false negative.

Reduce your risk of making Type II Error

One can reduce the risk of making a Type II error by ensuring that the test has enough power. In data work, power is usually set at 0.80 or 80%. The higher the statistical power, the lower the probability of making a Type II error. To increase power, you can increase your sample size or your significance level.

Potential Risks of Type I and Type II Errors

As a data professional, it is important to be aware of the potential risks involved in making the two types of errors.

  • A Type I error means rejecting a true null hypothesis. In general, making a Type I error often leads to implementing changes that are unnecessary and ineffective, and which waste valuable time and resources.
    For example, if you make a Type I error in your clinical trial, the new medicine will be considered effective even though it is ineffective. Based on this incorrect conclusion, ineffective medication may be prescribed to a large number of people. While other treatment options may be rejected in favor of the new medicine.
  • A Type II error means failing to reject a false null hypothesis. In general, making a Type II error may result in missed opportunities for positive change and innovation. A lack of innovation can be costly for people and organizations.
    For example, if you make a Type II error in your clinical trial, the new medicine will be considered ineffective even though it’s effective. This means that a useful medication may not reach a large number of people who could benefit from it.

In summary, as a data professional, it helps to be aware of the potential errors built into hypothesis testing and how they can affect the final decisions. Depending on the certain situation, one may choose to minimize the risk of either a Type I or Type II error. Ultimately, it is the responsibility of a data professional to determine which type of error is riskier based on the goals of your analysis.

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Easy Hypothesis Testing Questions MCQs 7

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Online MCQs Hypothesis Testing Questions with Answers. The Hypothesis Testing Quiz is important for the preparation of examinations and different statistical job tests in Government/ Semi-Government or Private Organization sectors. These tests are also helpful in getting admission to different colleges and Universities.

Online MCQs about Hypothesis Testing with Answers

1. Statistical inference is divided into two major branches as given below

 
 
 
 

2. For testing of the hypothesis about population proportion, we use

 
 
 
 

3. Which of the following cannot be considered as a hypothesis ($H_0$)

 
 
 
 

4. The alternative hypothesis may be of the form

 
 
 
 

5. The probability of accepting the true null hypothesis is called

 
 
 
 

6. The critical region is the part of the sampling distribution for which the null hypothesis is

 
 
 
 

7. If the alternative hypothesis is of the form, $H_1: \theta \ne \theta_0$ then the test will be

 
 
 
 

8. The hypothesis that is tested for the possible purpose of rejection is called

 
 
 
 

9. If a false hypothesis is accepted, it is called

 
 
 
 

10. Which of the following is not the composite hypothesis

 
 
 
 

11. Hypothesis testing and estimation are major classifications of

 
 
 
 

12. If the population standard deviation is not known and the sample size is large ($n\ge 30$) then the test statistic to be used is

 
 
 
 

13. The null hypothesis always contains some sign of

 
 
 
 

14. The ———— of a test is the maximum probability with which we are willing to a risk of Type-I error.

 
 
 
 

15. One-sided and two-sided tests are based on

 
 
 
 

16. In testing of the hypothesis, the sampling distribution of the test statistics is based on the assumption that

 
 
 
 

17. The probability of rejecting a true null hypothesis is called

 
 
 
 

18. If the alternative hypothesis is of the form $H_1:\theta < \theta_0$ then the test will be

 
 
 
 

19. The null hypothesis ($H_0$) is

 
 
 
 

20. In the t-test, we use

 
 
 
 

Question 1 of 20

Hypothesis Testing Questions MCQs

The goal of hypothesis testing is to assess the evidence against the null hypothesis (a statement about population also called claimed hypothesis). If the evidence is strong enough (based on the sample information), one can reject the null hypothesis and conclude that the alternative hypothesis is more likely to be true. However, it is important to remember that we never truly “prove” the alternative hypothesis; one can only reject the null hypothesis.

Hypothesis Testing Questions MCQs

  • Statistical inference is divided into two major branches as given below
  • The probability of accepting the true null hypothesis is called
  • The probability of rejecting a true null hypothesis is called
  • If a false hypothesis is accepted, it is called
  • Hypothesis testing and estimation are major classifications of
  • The hypothesis that is tested for the possible purpose of rejection is called
  • In testing of the hypothesis, the sampling distribution of the test statistics is based on the assumption that
  • Which of the following cannot be considered as a hypothesis ($H_0$)
  • The alternative hypothesis may be of the form
  • The null hypothesis always contains some sign of
  • Which of the following is not the composite hypothesis
  • The critical region is the part of the sampling distribution for which the null hypothesis is
  • The null hypothesis ($H_0$) is
  • One-sided and two-sided tests are based on
  • If the alternative hypothesis is of the form $H_1:\theta < \theta_0$ then the test will be
  • If the alternative hypothesis is of the form, $H_1: \theta \ne \theta_0$ then the test will be
  • If the population standard deviation is not known and the sample size is large ($n\ge 30$) then the test statistic to be used is
  • In the t-test, we use
  • For testing of the hypothesis about population proportion, we use
  • The ———— of a test is the maximum probability with which we are willing to a risk of Type-I error.
Hypothesis Testing Questions with Answers

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Inferential Statistics Tests List (2023)

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The following is the list of different parametric and non-parametric lists of the Inferential Statistics Tests List. A short description of each Inferential Statistics Test is also provided.

Inferential Statistics Tests: Parametric Statistics

Sr. No.Statistical TestShort Description of Inferential Test
1)Z testLarge sample test for one mean/average when sigma ($\sigma$) is known (or $n$ is large), population distribution is normal.
2)t testSmall sample test for one mean/average when sigma ($\sigma$) is unknown (and $n$ is small), population distribution is normal.
3)Z testLarge sample test for one proportion.
4)Z testSmall sample test for two means/averages when sigmas ($\sigma_1$ and $\sigma_2$) are unknown, samples are independent, and are from normal populations. The variances are NOT pooled.
5)t testSmall sample test for two means/averages when sigmas ($\sigma_1$ and $\sigma_2$) are unknown, samples are independent and are from normal populations. The variances are NOT pooled.
6)t testSmall sample test for two means/averages when sigmas ($\sigma_1$ and $\sigma_2$) are unknown, samples are independent and are from normal populations. The variances are NOT pooled.
7)t testA test for two means/averages for dependent (paired or related) samples where $d$ (The difference between samples) is normally distributed.
8)Z testLarge sample test for two proportions.
9)$\chi^2$Chi-square goodness of fit, or multinomial distribution., where each expected value is at least 5.
10)$\chi^2_{ii}$Chi-square for contingency tables (rows & columns) where each expected value is at least 5.
Either a test of independence, a test of homogeneity, or a test of association.
11)$\chi^2$Test for one variance or standard deviation.
12)F testTest for two variances or standard deviations for independent samples from the normal populations.
13)F (Anova)Test for three or more means for independent random samples from normal populations. The variances are assumed to be equal.
14)Tukey QA multiple comparison test for all pairs of means (usually for equal sample sizes).
15)Dunnett qA multiple comparison test for a control mean to other means.
16)Hartley H,
Bartlett,
Levene,
Brown-Forsythe,
O’Brien		Test for homoscedasticity, and homogeneity of variances.
17)Pearson $r$Pearson product-moment correlation coefficient.
18)SlopeTest on the slope of the linear regression line.
19)InterceptIntercept Test on the y-intercept of the linear regression line.

Inferential Statistics Tests: Non-Parametric Tests

The following is the list of non-parametric Tests with a short description of the tests.

Sr. No.Statistical TestsShort Description of Inferential Statistics Tests
1)Runs TestUsed to determine whether the sequence of data is random.
2)Mann-Whitney U TestAnalogous to Test # 5 from Parametric test list.
3)Sign TestAnalogous to Test # 2 from Parametric Test (Single sample Median Test). or Test # 7 from Parametric Test.
4)Wilcoxon Signed-Ranked TestSimilar to the sign test, but more efficient, analogous to Parametric Test # 7.
5)Kruskal-Wallis TestAnalogous to Parametric Test # 13.
6)Multiple Comparison TestAnalogous to Parametric Test # 14.
7)Spearmna $r_s$ Rank CorrelationAnalogous to Parametric Test # 15.

Advanced Inferential Statistics Tests

Following is the list of some advanced inferential Statistics tests with a short description of the test.

Sr. No. Statistical TestShort Description of Inferential Statistics Tests
1)Two-factor ANOVA With ReplicationInteraction is possible between two factors.
2)Two-factor ANOVA Without ReplicationOnly one observation per cell; no interaction effect is observed between the two factors.
3)One Datum StatUsed to compare one piece of data (datum) to a mean.
4)McNemar StAtUsed to test a 2 x 2 table of matched discordant pairs.
5)Two Poisson CountsUsed to compare two Poisson counts.
6)Two Regression SlopesUsed to compare two regression equation slopes.
7)Several Regression SlopesUsed to compare several regression equation slopes.
8)Multiple RegressionUsed to test a linear relationship with more than two variables.
9)Holgate StatisticUsed to determine spatial distribution.
Inferential Statistics Tests

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