Testing of Hypothesis

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. 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; large

 5/4; medium

 4/5; large

 4/5; medium

2. 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

 May be true or false

 False

 None of these

3. 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 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

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

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

4. 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.

 False

 It may be true or false

 True

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 ($\alpha = 0.05$).

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

 False

 True

 It may be true or false

7. 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.

 It may be true or false

 True

 False

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

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

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

9. 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 ($p = .001$).

 False

 True

 It may be true or false

10. 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 ————-.

 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

 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.

11. 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.

 It may be true or false

 True

 False

12. 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.

 It may be true or false

 True

 False

13. 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 ———–.

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

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

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

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

14. 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?

 80%

 20%

 80% * 80%

 20% * 20%

15. 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

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

 $1- (1 – \alpha)$ 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

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

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

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

 The same for all p-values

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

17. 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).

 True

 False

 It may by true or false

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

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

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

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

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

19. 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).

 True

 False

 It may be true or false

20. 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 $1-p$.

 It may be true or false

 False

 True

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Question 1 of 20

MCQs on Statistical Inference with Answers

Test your understanding of statistical hypothesis testing, Type I & Type II errors, p-values, significance levels, and research biases with this quiz. Key topics include:

• Type-I vs. Type-II errors and their rates
• Interpreting p-values (e.g., p = 0.30 vs. p = 0.001)
• Misconceptions about null & alternative hypotheses
• Statistical power, replication, and publication bias
• Pre-registration in research and its impact on results
• Effect size (Cohen’s d) and its interpretation

Perfect for students, researchers, and data analysts looking to strengthen their statistical reasoning and avoid common pitfalls in hypothesis testing.

• 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.

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R Programming Language and Statistics

In statistical hypotheses testing procedure, an important step is to determine whether to reject the null hypothesis. The step is to compute/find the critical values and rejection region.

Rejection Region and Critical Values

A rejection region for a hypothesis test is the range of values for the standardized test statistic which would lead us to decide whether to reject the null hypothesis. The Critical values for a hypothesis test are the z-scores which separate the rejection region(s) from the non-rejection region (also called the acceptance region of $H_0$).  The critical values will be denoted by $Z_0$.

The rejection region for a test is determined by the type of test (left-tailed, right-tailed, or two-tailed) and the level of significance (denoted by $\alpha$) for the test. For a left-tailed test, the rejection region is a region in the left tail of the normal distribution, for a right-tailed test, it is in the right tail, and for a two-tailed test, there are two equal rejection regions in either tail.

Once we establish the critical values and rejection region, if the standardized test statistics for a sample data set fall in the region of rejection, the null hypothesis is rejected.

Examples: Critical Values and Rejection Region

Example 1: A university claims that the average SAT score for its incoming freshmen is 1080. A sample of 56 freshmen at the university is drawn and the average SAT score is found to be $\overline{x}=1044$ with a sample standard deviation of $s=94.7$ points.

In the above SAT example, the test is two-tailed, so the rejection region will be the two tails at either end of the normal distribution. If we again want $\alpha=0.05$, then the area under the curve in both rejection regions together should be 0.05. For this purpose, we will look up $\frac{\alpha}{2}=0.025$ in the standard normal table to get critical values of $Z_0 = \pm 1.96$. The rejection region thus consists of $Z \le 1.96$ and $Z\ge 1.96$. Since the standardized test statistic $Z=-2.85$ falls in the region, the university’s claim of $\mu = 1080$ would be rejected in this case.

Example 2: Consider a left-tailed Z test. For a 0.05 level of significance, the rejection region would be the values in the lowest 5% of the standard normal distribution (5% lowest area under the normal curve). In this case, the critical value (the corresponding) Z-score will be $-1.645$. So the critical value $Z_0$ will be $-1.645$ and the rejection region will be $Z\le -1.645$.

Note that for the case of right-tailed the rejection region would be the values in the highest 5% of the standard normal distribution table. The Z-score will be $1.645$ and the rejection region will be $Z\ge 1.645$.

Exercise: Critical Values and Rejection Region

1. Find the critical values and rejection regions(s) for the standardized Z-test of the following:

• A right-tailed test with $\alpha = 0.05$
• A left-tailed test with $\alpha = 0.01$
• A two-tailed test with $\alpha = 0.10$
• A right-tailed test with $\alpha = 0.02$

2. Mercury levels in fish are considered dangerous to people if they exceed 0.5mg mercury per kilogram of meat. A sample of 50 tuna is collected, and the mean level of mercury in these 50 fishes is 0.6m/kg, with a standard deviation of 0.2mg/kg. A health warning will be issued if the claim that the mean exceeds 0.5mg/kg can be supported at the $\alpha=0.10$ level of significance. Determine the null and alternative hypotheses in this case, the type of the test, the critical value(s), and the rejection region. Find the standardized test statistics for the information given in the exercise. Should the health warning be issued?

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