Statistics for Data Analyst

Introduction to Types of Hypothesis Tests

In statistics, hypothesis tests are methods used to make inferences or draw conclusions about a population based on sample data. In this pose, we will discuss the Basic Types of Hypothesis Tests in Statistics. There are three basic types of hypothesis tests, namely (i) Left-Tailed Test, (ii) Right-Tailed Test, and (iii) Two-Tailed Test.

Note that I am not talking about Statistical tools used under specific conditions related to the data type and distribution. I am talking about the nature of the hypotheses being tested. Therefore, I will focus in this post on the area under the curve in the tails. In hypothesis testing, the distribution of the test’s rejection region can be characterized as either one-tailed or two-tailed. The one-tailed tests include both left- and right-tailed tests.

Left-Tailed Test

The left-tailed tests are used when the null hypothesis is being tested in a claim that the population parameter at least ($\ge$) a given value. Note that the alternative hypothesis then claims that the parameter is less than (<) the value. For example,

A tire manufacturer claims that their tires last on average more than 35000 miles. If one thinks that the claim is false, then one would write the claim as $H_o$, remembering to include the condition of equality. The hypothesis for this test would be: 
$$H_o:\mu\ge 35000$$
$$H_1: \mu<35000$$

One would hope that the sample data would allow the rejection of the null hypothesis, refuting the company’s claim.

The $H_o$ will be rejected in the case above if the sample mean is statistically significantly less than 35000. That is, if the sample mean is in the left-tail of the distribution of all sample means.

Right Tailed Test

The right-tailed test is used when the null hypothesis ($H_0$) being tested is a claim that the population parameter is at most ($\le$) a given value. Note that the alternative hypothesis ($H_1$) then claims that the parameter is greater than (>) the value.

Suppose, you worked for the tire company and wanted to gather evidence to support their claim then you will make the company's claim $H_1$ and remember that equality will not be included in the claim (H_o$). The hypothesis test will be

$$H_0:\mu \le 35000$$
$$H_1:\mu > 35000$$

If the sample data was able to support the rejection of $H_o$ this would be strong evidence to support the claim $H_1$ which is what the company believes to be true.

One should reject $H_o$ in this case if the sample mean was significantly more than 35000. That is, if the sample mean is in the right-tailed of the distribution of all sample means.

Two-Tailed Test

The two-tailed test is used when the null hypothesis ($H_o$ begins tested as a claim that the population parameter is equal to (=) a given value. Note that the alternative hypothesis ($H_1$) then claims that the parameter is not equal to ($\ne$) the value. For example, the Census Bureau claims that the percentage of Punjab’s area residents with a bachelor’s degree or higher is 24.4%. One may write the null and alternative hypotheses for this claim as:

$$H_o: P = 0.244$$
$$H_1: P \ne 0.244$$

In this case, one may reject $H_o$ if the sample percentage is either significantly more than 24.4% or significantly less than 24.4%. That is if the sample proportion was in either tail (both tails) of the distribution of all sample proportions.

Key Differences

• Directionality: One-tailed tests look for evidence of an effect in one specific direction, while two-tailed tests consider effects in both directions.
• Rejection Regions: In a one-tailed test, all of the rejection regions are in one tail of the distribution; in a two-tailed test, the rejection region is split between both tails.

Online Quiz and Test Website

R Programming Language and Frequently Asked Questions

The quiz is about MCQs on Statistical Inference with Answers. The quiz contains 20 questions about hypothesis testing and p-values. It covers the topics of formulation of the null and alternative hypotheses, level of significance, test statistics, region of rejection, and decision about acceptance and rejection of the hypothesis. Let us start with the Quiz MCQs on Statistical Inference.

Online MCQs on Statistical Inference with Answers

1. Suppose a research article indicates a $p = 0.001$ value in the results section ($\alpha = 0.05$).

The probability that the results of the given study are replicable is not equal to $1-p$.

 True

 False

 It may be true or false

2. After finding a single statistically significant p-value we can conclude that ————-, but it would be incorrect to conclude that ————.

 The data favors the the alternative hypothesis; the null hypothesis is likely to be false

 The data is surprising if we assume there is no effect; the null hypothesis is likely to be false

 The null hypothesis is likely to be false; the alternative hypothesis is true

 The data is surprising assuming the null hypothesis is true; the data is not surprising assuming the alternative hypothesis is true

3. When the null hypothesis is true, the probability of finding a specific p-value is ————-.

 Higher for lower p-values (e.g., 0.01) than for higher p-values (e.g., 0.04)

 The same for all p-values

 Higher for higher p-values (e.g., 0.04) than for lower p-values (e.g., 0.01)

 Equal to the p-value (e.g. a p of 0.06 has a 6% probability)

4. Person A is very skeptical about homeopathy. Person B believes strongly in homeopathy. They both read a study about homeopathy, which reports a positive effect and $p < 0.05$. Person A would be more likely than Person B to conclude that ———-, and Person B would be more likely than Person A to think that ————-.

 The study must be a Type 2 error; the null hypothesis that homeopathy is not true was correctly rejected

 The study might be a type 1 error; the hypothesis that homeopathy is not true was correctly rejected

 The study is likely a type 1 error; the study is likely a type 2 error

 The probability of homeopathy being true is high; the study provides evidence for the hypothesis that homeopathy has a true effect

5. Suppose a research article indicates a $p = 0.30$ value in the results section ($\alpha = 0.05$).

The alternative hypothesis has been shown to be false.

 False

 It may be true or false

 True

6. Suppose a research article indicates a $p = 0.30$ value in the results section ($\alpha = 0.05$).

You have found the probability of the null hypothesis being true ($p = 0.30$).

 True

 None of these

 False

 May be true or false

7. Suppose a research article indicates a value of $p = 0.30$ in the results section ($\alpha = 0.05$).

Obtaining a statistically non-significant result implies that the effect detected is unimportant.

 True

 False

 It may be true or false

8. Suppose that a research article indicates a value of $p = 0.001$ in the results section ($\alpha = 0.05$).

The null hypothesis has been shown to be false.

 True

 It may be true or false

 False

9. Suppose a research article indicates a $p = 0.001$ value in the results section ($\alpha = 0.05$).

You have absolutely proven your alternative hypothesis (that is, you have proven that there is a difference between the population means).

 It may be true or false

 True

 False

10. Suppose a research article indicates a $p = 0.30$ value in the results section ($\alpha = 0.05$).

You have proven the null hypothesis (that is, you have proven that there is no difference between the population means).

 False

 It may by true or false

 True

11. Suppose a research article indicates a value of $p = 0.001$ in the results section ($\alpha = 0.05$).

The p-value of a statistical test is the probability of the observed result or a more extreme result, assuming the null hypothesis is true.

 False

 True

 It may be true or false

12. You perform two studies to test a potentially life-saving drug. Both studies have 80% power. What is the chance of two type 2 errors (of false negatives) in a row?

 20% * 20%

 80%

 20%

 80% * 80%

13. Suppose a research article indicates a value of $p = 0.001$ in the results section ($\alpha = 0.05$).

You have found the probability of the null hypothesis being true ($p = .001$).

 True

 False

 It may be true or false

14. When the difference between means is 5, and the standard deviation is 4, Cohen’s d is ————— which is ————— according to the benchmarks proposed by Cohen.

 5/4; medium

 5/4; large

 4/5; large

 4/5; medium

15. Suppose a research article indicates a $p = 0.30$ value in the results section ($\alpha = 0.05$).

The p-value gives the probability of obtaining a significant result whenever a given experiment is replicated.

 True

 False

 It may be true or false

16. It is important to have access to all (and not just statistically significant) research findings to be able to ————. A consequence of publication bias is that ———–.

 Reliably estimate effect sizes; you cannot reliably quantify the size of an effect

 Reliably estimate effect sizes; the type 1 error rate is inflated

 Estimate the overall error rates in the literature; you cannot reliably generate theories

 Generate theories; you cannot reliably quantify the size of an effect

17. Study A and B are completely identical, except that all tests reported in Study A were pre-registered at a publicly available location (and the reported tests match the pre-registered tests), but all tests in Study B are not pre-registered. Both contain analyses with covariates. Based on research on flexibility in the data analysis, we can expect that on average study A will have ————; the covariate analyses are ————-.

 The same type 1 error rate as Study B; problematic for both studies

 An inflated Type 1 error rate due to multiple comparisons; problematic for the not pre-registered study, because they always inflate the Type 1 error rate unless pre-registered.

 A controlled Type 1 error rate; not problematic for the pre-registered study, and perhaps (but not necessarily) inflating the Type 1 error rate in the not pre-registered study.

 An inflated Type 1 error rate due to multiple comparisons; problematic for both studies as covariates always inflate the Type 1 error rate

18. Suppose a research article indicates a $p = 0.001$ value in the results section ($\alpha = 0.05$).

Obtaining a statistically significant result implies that the effect detected is important.

 True

 It may be true or false

 False

19. When $H_0$ is true, the probability that at least 1 out of a $X$ completely independent findings is a Type 1 error is equal to ————, this probability ———— when you look at your data and collect more data if a test is not significant.

 $1- (1 – \alpha) \times X$; remains equal because the alpha decides the error rate of individual studies

 $1- (1 – \alpha)$ to the power of $X$; increases, unless you perform a correction

 $1- (1 -\alpha)$ to the power of $X$; remains equal because the alpha decides the error rate of individual studies

 The $\alpha \times X$; increases, unless you perform a correction

20. A Type-I error is ————–, and the Type-I error rate is determined by ————–.

 Finding a significant result, when there is a true effect; the power of your study

 Finding a significant result, when there is a no true effect; the alpha level

 Drawing an incorrect conclusion about the present or absence of an effect; the power of your study

 Not finding a significant result, when there is a true effect; the alpha level

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MCQs on Statistical Inference with Answers

• A Type-I error is ————–, and the Type-I error rate is determined by ————–.
• Suppose a research article indicates a $p = 0.30$ value in the results section ($\alpha = 0.05$). You have found the probability of the null hypothesis being true ($p = 0.30$).
• Suppose a research article indicates a $p = 0.30$ value in the results section ($\alpha = 0.05$). You have proven the null hypothesis (that is, you have proven that there is no difference between the population means).
• Suppose that a research article indicates a value of $p = 0.001$ in the results section ($\alpha = 0.05$). The null hypothesis has been shown to be false.
• Suppose a research article indicates a $p = 0.30$ value in the results section ($\alpha = 0.05$). The p-value gives the probability of obtaining a significant result whenever a given experiment is replicated.
• Suppose a research article indicates a value of $p = 0.30$ in the results section ($\alpha = 0.05$). Obtaining a statistically non-significant result implies that the effect detected is unimportant.
• Suppose a research article indicates a value of $p = 0.001$ in the results section ($\alpha = 0.05$). The p-value of a statistical test is the probability of the observed result or a more extreme result, assuming the null hypothesis is true.
• Suppose a research article indicates a $p = 0.001$ value in the results section ($\alpha = 0.05$). Obtaining a statistically significant result implies that the effect detected is important.
• Suppose a research article indicates a $p = 0.001$ value in the results section ($\alpha = 0.05$). You have absolutely proven your alternative hypothesis (that is, you have proven that there is a difference between the population means).
• Suppose a research article indicates a value of $p = 0.001$ in the results section ($\alpha = 0.05$). You have found the probability of the null hypothesis being true ($p = .001$).
• Suppose a research article indicates a $p = 0.001$ value in the results section ($\alpha = 0.05$). The probability that the results of the given study are replicable is not equal to $1-p$.
• Person A is very skeptical about homeopathy. Person B believes strongly in homeopathy. They both read a study about homeopathy, which reports a positive effect and $p < 0.05$. Person A would be more likely than Person B to conclude that ———-, and Person B would be more likely than Person A to think that ————-.
• You perform two studies to test a potentially life-saving drug. Both studies have 80% power. What is the chance of two type 2 errors (of false negatives) in a row?
• Study A and B are completely identical, except that all tests reported in Study A were pre-registered at a publicly available location (and the reported tests match the pre-registered tests), but all tests in Study B are not pre-registered. Both contain analyses with covariates. Based on research on flexibility in the data analysis, we can expect that on average study A will have ————; the covariate analyses are ————-.
• When the null hypothesis is true, the probability of finding a specific p-value is ————-.
• After finding a single statistically significant p-value we can conclude that ————-, but it would be incorrect to conclude that ————.
• When $H_0$ is true, the probability that at least 1 out of a $X$ completely independent findings is a Type 1 error is equal to ————, this probability ———— when you look at your data and collect more data if a test is not significant.
• It is important to have access to all (and not just statistically significant) research findings to be able to ————. A consequence of publication bias is that ———–.
• When the difference between means is 5, and the standard deviation is 4, Cohen’s d is ————— which is ————— according to the benchmarks proposed by Cohen.
• Suppose a research article indicates a $p = 0.30$ value in the results section ($\alpha = 0.05$). The alternative hypothesis has been shown to be false.

Online MCQs and Test Website, gmstat.com

R Programming Language and Statistics