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Standard Error from a Formula and a Bootstrap Distribution Sample A has a count of 30 successes with and Sample B has a count of 50 successes with . Use StatKey or other technology to generate a bootstrap distribution of sample differences in proportions and find the standard error for that distribution. Compare the result to the value obtained using the formula for the standard error of a difference in proportions from this section.

Respuesta :

MrRoyal

Answer:

Using a formula, the standard error is: 0.052

Using bootstrap, the standard error is: 0.050

Comparison:

The calculated standard error using the formula is greater than the standard error using bootstrap

Step-by-step explanation:

Given

Sample A                          Sample B

[tex]x_A = 30[/tex]                              [tex]x_B = 50[/tex]

[tex]n_A = 100[/tex]                             [tex]n_B =250[/tex]

Solving (a): Standard error using formula

First, calculate the proportion of A

[tex]p_A = \frac{x_A}{n_A}[/tex]

[tex]p_A = \frac{30}{100}[/tex]

[tex]p_A = 0.30[/tex]

The proportion of B

[tex]p_B = \frac{x_B}{n_B}[/tex]

[tex]p_B = \frac{50}{250}[/tex]

[tex]p_B = 0.20[/tex]

The standard error is:

[tex]SE_{p_A-p_B} = \sqrt{\frac{p_A * (1 - p_A)}{n_A} + \frac{p_A * (1 - p_B)}{n_B}}[/tex]

[tex]SE_{p_A-p_B} = \sqrt{\frac{0.30 * (1 - 0.30)}{100} + \frac{0.20* (1 - 0.20)}{250}}[/tex]

[tex]SE_{p_A-p_B} = \sqrt{\frac{0.30 * 0.70}{100} + \frac{0.20* 0.80}{250}}[/tex]

[tex]SE_{p_A-p_B} = \sqrt{\frac{0.21}{100} + \frac{0.16}{250}}[/tex]

[tex]SE_{p_A-p_B} = \sqrt{0.0021+ 0.00064}[/tex]

[tex]SE_{p_A-p_B} = \sqrt{0.00274}[/tex]

[tex]SE_{p_A-p_B} = 0.052[/tex]

Solving (a): Standard error using bootstrapping.

Following the below steps.

  • Open Statkey
  • Under Randomization Hypothesis Tests, select Test for Difference in Proportions
  • Click on Edit data, enter the appropriate data
  • Click on ok to generate samples
  • Click on Generate 1000 samples ---- see attachment for the generated data

From the randomization sample, we have:

Sample A                          Sample B

[tex]x_A = 23[/tex]                              [tex]x_B = 57[/tex]

[tex]n_A = 100[/tex]                             [tex]n_B =250[/tex]

[tex]p_A = 0.230[/tex]                          [tex]p_A = 0.228[/tex]

So, we have:

[tex]SE_{p_A-p_B} = \sqrt{\frac{p_A * (1 - p_A)}{n_A} + \frac{p_A * (1 - p_B)}{n_B}}[/tex]

[tex]SE_{p_A-p_B} = \sqrt{\frac{0.23 * (1 - 0.23)}{100} + \frac{0.228* (1 - 0.228)}{250}}[/tex]

[tex]SE_{p_A-p_B} = \sqrt{\frac{0.1771}{100} + \frac{0.176016}{250}}[/tex]

[tex]SE_{p_A-p_B} = \sqrt{0.001771 + 0.000704064}[/tex]

[tex]SE_{p_A-p_B} = \sqrt{0.002475064}[/tex]

[tex]SE_{p_A-p_B} = 0.050[/tex]

Ver imagen MrRoyal