Sampling with Replacement

In sampling with replacement, the units drawn are returned to the population before drawing the next unit. This means the same individual can be chosen more than once in the sampling process. The sampling with replacement may provide valuable insights while maintaining flexibility in selecting samples from a given population.

Key Characteristics of Sampling with Replacement

The following are key characteristics of Sampling with Replacement:

  1. Independence: Each selection is independent, as the same item can be selected multiple times.
  2. Population Size: The effective population size remains the same for each draw since previously selected items are replaced.
  3. Use Cases: This method is commonly used in algorithms, simulations, and bootstrapping techniques in statistics, where it’s important to assess variability or make inferences from a sample.

Example of Sampling with Replacement

As an example of sampling with replacement, suppose, you have a bag containing three colored balls (red, blue, and green), and you sample with a replacement, if you draw a red ball, you put it back into the bag before the next draw. As a result, in subsequent draws, you could again draw a red ball.

Drawing All Possible Samples Using Sampling with Replacement

Question: Consider a population with elements A, B, C, and D. Draw all possible samples of size 2 with replacement from this population.

Solution: In this problem, $N=4$ and $n=2$.

Possible number of samples (with replacement) = $N^n = 4^2 = 16$.

The 16 samples of size 2 are

AAABACAD
BABBBCBD
CACBCCCD
DADBDCDD

Question: Draw all possible samples of size 3 with replacement from a population having elements 2, 4, and 6.

Solution:

Population size = $N=3$, Sample size = n = 3$

Number of possible samples are $N^n = 3^3 = 27$

There are two ways to list these samples.

First Method:

First divide possible samples (27) by the population size unit quotient 1 is returned. For example, $\frac{27}{3} = 9, \quad \frac{9}{3}, \quad \frac{9}{3}=1$.

We obtained three quotients: 9, 3, and 1. These are the number of repetitions of population units. First, write every unit 9 times, then 3 times, and lastly, write every unit 1 time.

Sampling with Replacement

Second Method:

First, make the samples of size 2, which are easy to draw.

2, 2
2, 4
2, 6
4, 2
4, 4
4, 6
6, 2
6, 4
6, 6

Repeat these samples three times. Since the required number of samples is 27, add every population unit at (the start or) at the end of these samples of size two.

2, 2, 22, 2, 42, 2, 6
2, 4, 22, 4, 42, 4, 6
2, 6, 22, 6, 42, 6, 6
4, 2, 24, 2, 44, 2, 6
4, 4, 24, 4, 44, 4, 6
4, 6, 24, 6, 44, 6, 6
6, 2, 26, 2, 46, 2, 6
6, 4, 26, 4, 46, 4, 6
6, 6, 26, 6, 46, 6, 6

From the table above, 2 is added in the last of the first nine samples, then 4 is added in the last of the next 9 samples and finally 6 is added in the last nine samples.

Real-Life Examples of Sampling with Replacement

The following are some real-life examples of sampling with replacement:

  1. Lottery Draws: In some types of lotteries, numbers can be drawn multiple times before the final selection. For example, if a lottery allows for the same number to be drawn again after being selected, this is akin to sampling with replacement.
  2. Quality Control in Manufacturing: In a factory, inspectors might draw samples of products to test for defects. After testing each item, they return it to the production line before drawing the next sample to maintain the same population size and ensure each product has a chance of being selected again.
  3. Genetic Studies: In genetics, researchers might take DNA samples from a population to study traits or disorders. By replacing each sample with the population (considering genetic diversity), they can analyze the data while allowing for the possibility of selecting the same individual multiple times.
  4. Surveys: When conducting surveys, researchers might randomly select participants from a population (like voters or consumers) and, after querying each individual, they can include them again in the pool for subsequent selections, especially in larger datasets where the same individuals might provide valuable insights if repeated.
  5. Educational Testing: In standardized testing, students might take multiple attempts at a test where scores from previous attempts can be considered again in analyses to assess trends in learning or improvement.
  6. Customer Behavior Analysis: Companies may analyze customer purchase patterns by repeatedly sampling transactions. For instance, if a customer makes multiple purchases, their transaction data might be included in each analysis to understand their buying behavior over time.

Sampling Quiz Questions

Simulation and Sampling in R

Hypothesis Testing MCQs 10

The quiz is about Hypothesis Testing MCQs 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, decision, effect size, about acceptance and rejection of the hypothesis. Let us start with the Quiz Hypothesis Testing MCQs Quiz now.

Online Hypothesis Testing MCQs with Answers

Online Hypothesis Testing MCQs with Answers

1. Consider a normally distributed data set with mean $\mu = 63.18$ inches and standard deviation $\sigma = 13.27$ inches. What is the z-score when $x = 91.54$ inches?

 
 
 
 

2. What is the purpose of an ANOVA test?

 
 
 
 

3. Going through a dataset and looking at which effects are present can be problematic when —————-. It is NOT problematic when you ————–.

 
 
 
 

4. Predicting that a measured variable differs in two groups, without random assignment to conditions, is often ——————.

 
 
 
 

5. An experiment has been conducted to test the equality of two means, with known variances. The P-value for the Z-test statistic was 0.023. Assume a two-sided alternative hypothesis. The 95% confidence interval on the difference in the two means included the value zero.

 
 

6. An example of an unstandardized effect size is ——————; unstandardized effect sizes ——————.

 
 
 
 

7. You replicate an older study, which reported both credible intervals and confidence intervals. You also calculate both. Which statement is correct?

 
 
 
 

8. Which of the following statements about the ANOVA F-test score are true?

 
 
 
 

9. The battery life of smartphones is of great concern to customers. A consumer group tested four brands of smartphones to determine the battery life. Samples of phones of each brand were fully charged and left to run until the battery died. The table above displays the number of hours each of the batteries lasted. What test will be used to test the difference in means?

 
 
 
 

10. You perform five tests without correcting for multiple comparisons. The error rate for each test is ————–. After using the Bonferonni correction, the individual error rate for each test is —————.

 
 
 
 

11. A room in a laboratory is only considered safe if the mean radiation level is 400 or less. When a sample of 10 radiation measurements was taken, the mean value of the radiation was 414 with a standard deviation of 17. Some concerns mean radiation is above 414. Radiation levels in the lab are known to follow a normal distribution with a standard deviation of 22. We would like to conduct a hypothesis test at the 5% level of significance to determine whether there is evidence that the laboratory is unsafe. What will be the appropriate test?

 
 
 
 

12. The difference between eta-squared and partial eta-squared is ————, the difference between eta-squared and omega-squared is ————–

 
 
 
 

13. An experiment has been performed with a factor having two levels. There are 10 observations at each level. The following data results:
$\overline{y_1} = 10.5, S_1=2, \overline{y_2}=12.4, S_2=1.6$
You conduct a test of the hypothesis that the two means are equal. Assume that the alternative hypothesis is two-sided and that the population variances are equal. The P-value is:

 
 
 
 

14. Using the teacher’s rating data, is there an association between native (native English speakers) and the number of credits taught? What test will you use?

 
 
 
 

15. You performed a p-curve analysis and found a skewed distribution of p-values which peaks around $p = 0.045$, what does this mean?

 
 
 
 

16. If I wanted to test for association using a chi-square test, whether there is an association between gender (Male or Female) and tenure-ship (tenured or not tenured), what would be my degree of freedom?

 
 
 
 

17. The most important assumption in using the t-test is that the sample data come from normal populations.

 
 

18. You predict that your intervention will increase all participants’ performance on a test, this is an example of —————–. After the study, you conclude that the intervention only works for women but not men, this is an example of —————–.

 
 
 
 

19. In studies with less observations, parameters like effect sizes vary ______, the power to detect the effect size in the population depends, among other things, on _____.

 

 
 
 
 

20. Which of the following is a possible alternative hypothesis $H_1$ for a two-tailed test?

 
 
 
 

Online Hypothesis Testing MCQs with Answers

  • You perform five tests without correcting for multiple comparisons. The error rate for each test is ————–. After using the Bonferonni correction, the individual error rate for each test is —————.
  • An example of an unstandardized effect size is ——————; unstandardized effect sizes ——————.
  • The difference between eta-squared and partial eta-squared is ————, the difference between eta-squared and omega-squared is ————–
  • You replicate an older study, which reported both credible intervals and confidence intervals. You also calculate both. Which statement is correct?
  • In studies with less observations, parameters like effect sizes vary —————, the power to detect the effect size in the population depends, among other things, on —————–.  
  • You performed a p-curve analysis and found a skewed distribution of p-values which peaks around $p = 0.045$, what does this mean?
  • You predict that your intervention will increase all participants’ performance on a test, this is an example of —————–. After the study, you conclude that the intervention only works for women but not men, this is an example of —————–.
  • Predicting that a measured variable differs in two groups, without random assignment to conditions, is often ——————.
  • Going through a dataset and looking at which effects are present can be problematic when —————-. It is NOT problematic when you ————–.
  • What is the purpose of an ANOVA test?
  • Which of the following is a possible alternative hypothesis $H_1$ for a two-tailed test?
  • Using the teacher’s rating data, is there an association between native (native English speakers) and the number of credits taught? What test will you use?
  • If I wanted to test for association using a chi-square test, whether there is an association between gender (Male or Female) and tenure-ship (tenured or not tenured), what would be my degree of freedom?
  • Consider a normally distributed data set with mean $\mu = 63.18$ inches and standard deviation $\sigma = 13.27$ inches. What is the z-score when $x = 91.54$ inches?
  • The battery life of smartphones is of great concern to customers. A consumer group tested four brands of smartphones to determine the battery life. Samples of phones of each brand were fully charged and left to run until the battery died. The table above displays the number of hours each of the batteries lasted. What test will be used to test the difference in means?
  • A room in a laboratory is only considered safe if the mean radiation level is 400 or less. When a sample of 10 radiation measurements was taken, the mean value of the radiation was 414 with a standard deviation of 17. Some concerns mean radiation is above 414. Radiation levels in the lab are known to follow a normal distribution with a standard deviation of 22. We would like to conduct a hypothesis test at the 5% level of significance to determine whether there is evidence that the laboratory is unsafe. What will be the appropriate test?
  • Which of the following statements about the ANOVA F-test score are true?
  • An experiment has been performed with a factor having two levels. There are 10 observations at each level. The following data results: $\overline{y_1} = 10.5, S_1=2, \overline{y_2}=12.4, S_2=1.6$ You conduct a test of the hypothesis that the two means are equal. Assume that the alternative hypothesis is two-sided and that the population variances are equal. The P-value is:
  • An experiment has been conducted to test the equality of two means, with known variances. The P-value for the Z-test statistic was 0.023. Assume a two-sided alternative hypothesis. The 95% confidence interval on the difference in the two means included the value zero.
  • The most important assumption in using the t-test is that the sample data come from normal populations.

R Language and Data Analysis

Probability Distribution Quiz 8

The post is about the MCQs Probability Distributions Quiz. There are 20 multiple-choice questions about probability distributions covering distributions such as discrete and continuous Binomial Probability Distribution, Bernoulli Probability Distribution, Poisson Probability Distribution, Poisson Probability, Distribution, Geometric Probability Distribution, Hypergeometric Probability Distribution, Chi-Square distribution, Normal distribution, and F-distribution. Let us start with the MCQs Discrete Probability Distributions Quiz.

MCQs Probability Distribution Quiz

Please go to Probability Distribution Quiz 8 to view the test

Online Probability Distribution Quiz

  • You find a z-score of -1.99. Which statement(s) is/are true?
  • Expected values are properties of what?
  • If you got a 75 on a test in a class with a mean score of 85 and a standard deviation of 5, the z-score of your test score would be
  • The spread of the normal curve depends upon the value of:
  • Which of the following can best be described as a normal distribution?
  • In its standardized form, the normal distribution
  • A test is administered annually. The test has a mean score of 150 and a standard deviation 20. If Chioma’s z-score is 1.50, what was her score on the test?
  • The P-value for a normally distributed right-tailed test is P=0.042. Which of the following is INCORRECT?
  • The time X taken by a cashier in a grocery store express lane to complete a transaction follows a normal distribution with a mean of 90 seconds and a standard deviation of 20 seconds. What is the first quartile of the distribution of X (in seconds)?
  • Green sea turtles have normally distributed weights, measured in kilograms, with a mean of 134.5 and a variance of 49.0. A particular green sea turtle’s weight has a z-score of -2.4. What is the weight of this green sea turtle? Round to the nearest whole number.  
  • We look for a model, as realistic as possible, for a continuous random variable $X$ that represents the lifetime of a machine, and whose mean and variance are equal to 1 and 3, respectively. Which of the following distributions can be acceptable?
    Uniform
    Exponential
    Gamma
    Gaussian
  • The square of a Gaussian N(1, 3)
  • The distribution function of the random variable $X$ is given by $F_X(x)=1-\frac{1}{x^2}$ for $x \ge c$, 0 otherwise, where $c$ is a constant. What is the set of possible values of the constant $c$?
  • A random variable $Y$ has the following distribution y:     -1   0   1    2 p(y):  3C 2C 0.4 0.1 The value of the constant C is
  • If $Z$ has a standard normal distribution, if $U$ has a chi-square distribution with $k$ degrees of freedom and if $Z$ and $U$ are independent then the distribution of $X=\frac{Z}{\sqrt{\frac{U}{\sqrt{k}}}}$ is
  • If $X$ is a F-distributed random variable with $m$ and $n$ df, then $W=\frac{mX/n}{1+mX/n}$ has a
  • The number of parameters in multivariate normal distribution having $p$ variables are
  • The moment generating function of Gamma distribution with parameter $\lambda$ and $k$ is
  • The moment generating function of normal distribution is
  • When the experiment is repeated a variable number of times to obtain a fixed number of successes is
  • If the mean of the Chi-Square distribution is 4 then its variance is

MCQs General Knowledge