Option 1: Replicable simulation given a seed value

When you press the Update button the first time, it will draw a random sample based on the INPUT values. The random sample will be drawn given the seed value included in the text box for seed value (default is given as 12345).

In the next step, you may wish to alter the inputs again OR keep the inputs unaltered and press the Update button again. This will conduct a second random sampling conditional on the given INPUT values.

For example, reload the app for Normal distribution. Keep all the INPUT values at its default values including the seed value as 12345. Click the Update button THREE times. Note the Sample Mean and SD displayed in the plot on the right (see the app). Three successive values displayed should be as follows:

• On the first click of ‘Update’: \(\bar{x} = 20.07\), \(s = 4.55\)
• On the second click of ‘Update’: \(\bar{x} = 18.13\), \(s = 3.88\)
• on the third click of ‘Update’: \(\bar{x} = 19.43\), \(s = 4.09\)

Since we generated the sample from a pre-specified seed value (here 12345, but you can change it to any value), these outcomes are replicable. That means if you reload the app and do it again, you will get the identical three values of \(\bar{x}\) and \(s\) of the sample in successive three clicks. In other words, if you and your friend run the app together with the identical seed value and other input values, both will get identical estimates.

Option 2: Non-replicable instant simulation

If you wish to explore many scenarios and do not necessarily wish to replicate your outputs, then select the option: Check the box to update instantly. The important difference here is that you cannot obtain multiple outcomes for the same input values as in Option 1. You have to change the input values (at least a minor change) to trigger the app to sample data again. In other words, the app in this option only ‘reacts’ when you change the INPUT values. Also, note that the Update button will not respond in this option and the random sampling will not be conducted using the seed value (as in Option 1). In Option 2, the seed value is instantly set based on the current time and process id, therefore, outputs are not replicable.

Normal Distribution

• Simulation Features
  • Check the box to update instantly
  • Update
  • Seed value for generating the random number
• Population Parameters
  • True Population Mean: \(\mu\)
  • True Population Standard Deviation: \(\sigma\)
  • X-axis scale for the center and scale effect
• Sample Characteristics
  • Sample: Number of observations
  • Number of bins
  • Plot type: Frequency Distribution; Overlay Normal Density
• Distribution Function
  • Cumulative probability
  • Probability Tail

Sampling Distribution

• Simulation Features
  • Check the box to update instantly
  • Update
  • Seed value for generating the random number
• Population Parameters
  • True Population Mean: \(\mu\)
  • True Population Standard Deviation: \(\sigma\)
• Sample Characteristics
  • Sample size

Hypothesis Testing: One-Sample Z-Test

• Simulation Features
  • Check the box to update instantly
  • Update
  • Seed value for generating the random number
• Population Parameters
  • True Population Mean: \(\mu\)
  • Hypothesised Population Mean: \(\mu_0\)
  • True Population Standard Deviation: \(\sigma\)
• Sample Characteristics
  • Sample size
• Distribution Function
  • Type 1 Error
  • Probability Tail: Lower tail; Upper tail; Both tails

Hypothesis Testing: One-Sample Student’s t-Test

• Simulation Features
  • Check the box to update instantly
  • Update
  • Seed value for generating the random number
• Population Parameters
  • True Population Mean: \(\mu\)
  • Hypothesised Population Mean: \(\mu_0\)
  • True Population Standard Deviation: \(\sigma\)
• Sample Characteristics
  • Sample size
• Distribution Function
  • Type 1 Error
  • Probability Tail: Lower tail; Upper tail; Both tails

Hypothesis Testing: Two-Sample Independent (Unpaired) t-Test

• Simulation Features
  • Check the box to update instantly
  • Update
  • Seed value for generating the random number
• Population Parameters
  • True Population 1 Mean: \(\mu_1\)
  • True Population 2 Mean: \(\mu_2\)
  • True Population Standard Deviation: \(\sigma\)
• Sample Characteristics
  • Sample 1 (Group 1) Size: \(n_1\)
  • Sample 2 (Group 2) Size: \(n_2\)
• Distribution Function
  • Type 1 Error
  • Probability Tail: Lower (Left) tail; Upper (Right) tail; Both (Two) tails

Hypothesis Testing: One-way Analysis of Variance

• Population Parameters
  • True Population 1 Mean: \(\mu_1\)
  • True Population 2 Mean: \(\mu_2\)
  • True Population 3 Mean: \(\mu_3\)
  • True Population Standard Deviation: \(\sigma\)
• Sample Characteristics
  • Sample 1 (Group 1) Size: \(n_1\)
  • Sample 2 (Group 2) Size: \(n_2\)
  • Sample 3 (Group 3) Size: \(n_3\)
• Distribution Function
  • Type 1 Error
  • Probability Tail: Lower (Left) tail; Upper (Right) tail; Both (Two) tails
• Simulation Features
  • Check the box to update instantly
  • Update
  • Seed value for generating the random number

Normal Distribution

• Sample
• Distribution
• Probability & Quantile

Sampling Distribution

• Population & Sample
• Sample
• Sample Estimators
• Confidence Interval
• Summary

Hypothesis Testing: One-Sample Z-Test

• Population
• Sample
• Test Statistic
• Summary

Hypothesis Testing: One-Sample Student’s t-Test

• Population
• Sample
• Test Statistic
• Summary

Hypothesis Testing: Two-Sample Independent (Unpaired) t-Test

• Population
• Sample
• Test Statistic
• Summary

Hypothesis Testing: One-way Analysis of Variance

• Population
• Sample
• SS & MS (Sum of Squares & Mean Squares)
• Test Statistic
• Summary