MCQs on Statistical Inference 9

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. 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 ————-.

 
 
 
 

2. 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).

 
 
 

3. 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.

 
 
 
 

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.

 
 
 

5. 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 ($p = .001$).

 
 
 

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 ($p = 0.30$).

 
 
 
 

7. 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.

 
 
 

8. 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.

 
 
 

9. 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).

 
 
 

10. 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 $1-p$.

 
 
 

11. 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.

 
 
 

12. 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.

 
 
 

13. 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?

 
 
 
 

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

 
 
 
 

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

 
 
 
 

16. 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.

 
 
 

17. 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 ———–.

 
 
 
 

18. 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.

 
 
 
 

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

 
 
 
 

20. 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 ————-.

 
 
 
 

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.
MCQs on Statistical Inference

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R Programming Language and Statistics

One Sample Hypothesis Test (t-test)

Introduction: One Sample Hypothesis Test

In this post, I will discuss One Sample Hypothesis Test (One Sample t-test). When testing a claim about the mean using sample data with a small number of observations (i.e., sample size), the appropriate t-distribution instead of the standard normal distribution should be used to determine the standardized test statistic, critical values, rejection region, and p-values.

Recall that if the sample of values drawn follows the normal distribution, the sample size (number of observations in the sample) is less than 30, and the population standard deviation is unknown, then the random variable

$$t=\frac{\overline{x} – \mu}{\frac{s}{\sqrt{n}}}$$

has the Student’s t-distribution with $n-1$ degrees of freedom.

The procedure of locating the rejection regions for a t-distribution hypotheses test is the same as for the normal distribution tests, however, the critical values will differ. To find the critical value(s) $t_0$ for a test, determine if the test is one-tailed or two-tailed and the significance level ($\alpha$). The critical values can be found in the t-distribution table by looking up the entry in the column giving the level of significance and the row showing the degrees of freedom.

Note that:

  • For a right-tailed test, $t_0$ is the positive value in the table
  • For a left-tailed test, $t_0$ is the negative of the value in the table
  • For a two-tailed test, there are two critical values $t_0$ both the value and its opposite.
One Sample Hypothesis Testing-Tails-Critical-Region

Assumptions of the One Sample Hypothesis Test (t-test)

  • Independence: Observations in the sample should be independent of each other.
  • Normality: The population from which the sample is drawn should be normally distributed. However, the t-test is relatively robust to violations of normality, especially for larger sample sizes.
  • Random Sampling: The sample should be a random sample from the population.

One Sample Hypothesis Test for Mean

Example 1: SAT Math scores are normally distributed. A sample of SAT Math scores for 16 students has an average score of 522.8 with a sample standard deviation of 154.5. Suppose, one wishes to support the claim that the average SAT Math score exceeds 500 using a level of significance of 0.05.

Solution

Step 1: The null and alternative hypotheses test in this case are

$H_0: \mu \le 500$ vs $H_1: \mu > 500$

From the alternative hypothesis, the test is right-hand-tailed with $\mu_0=500$.

Step 2: Level of Significance is 5% = 0.05

Step 3: Critical Value

Using the t-distribution table with one Tail, 5% level of significance, and $n-1=16-1=15$ degrees of freedom, the critical value is $t_0=1.753$. The rejection region is thus $t\ge 1.7535$.

Step 4: Test Statistics

The standardized test statistic is

\begin{align*}
t&=\frac{\overline{X} – \mu_0 }{\frac{s}{\sqrt{n}}}\\
&= \frac{522.8 – 500}{\frac{154.5}{\sqrt{16}}}\\
&= \frac{22.8}{38.625} = 0.59
\end{align*}

Step 5: Interpretation of One Sample Mean Test

Since the standardized test statistic is not in the region of rejection, therefore, one should not reject $H_0$ and so the sample data is not sufficient to support the claim that the average exceeds 500 at the 0.05 level of significance.

Example 2: A biologist measures the weights of anesthetized female grizzly bears during winter. A sample of 14 bears is found to have an average weight of $\overline{X} = 376.6$lbs. with a sample standard deviation of $s=32.5$lbs. Is there sufficient evidence to support the claim that the average weight of all female bears in the area is less than 400 lbs? Use $\alpha=0.01$ level of significance.

Solution:

Step 1: The null and alternative hypotheses in this case

$H_0:\mu \ge 400$ vs $H_1:\mu < 400$

From the alternative hypothesis, the test is left-hand-tailed with $\mu_0=400$.

Step 2: Level of Significance is 1% = 0.01

Step 3: Critical Value

Using the Student’s t-distribution table with one tail, the level of significance $\alpha = 0.01$, and $n-1=14-1=13$ degrees of freedom, the critical value $t_0=-2.650$. The region of rejection is thus $t\le -2.650$.

Step 4: The Test Statistics is

\begin{align*}
t&=\frac{\overline{X} – \mu_0 }{\frac{s}{\sqrt{n}}}\\
&= \frac{376.6 – 400}{\frac{32.5}{\sqrt{14}}}\\
&= \frac{-23.4}{8.686} = -2.694
\end{align*}

Since the standardized test statistic is in the rejection region, one should reject the null hypothesis ($H_0$), which supports the claim that the average bear weight is less than 400 lbs.

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Critical Values and Rejection Region

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.

Hypothesis-Testing-Tails-Critical Values and Rejection Region

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 $Z\ge 1.645$.

    Hypothesis Test

    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 $\alpha = 0.02$
    1. 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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