Consider the following sample of three measurements {4,8,13}.Five bootstrap samples of this sample produced the following resamples: {4,8,8},{4,4,13},{4,8,13},{8,8,8},{8,13,13}.Based on these five resamples,find the bootstrap of the standard error of the sample mean.
A)3.4111
B)1.8469
C)2.3157
D)0.8260
E)8.2667

Question

Quizplus · Accepted Answer

To find the bootstrap estimate of the standard error of the sample mean, we first calculate the mean of each bootstrap sample, then calculate the standard deviation of these means. Means of the bootstrap samples:- {4,8,8} = (4+8+8)/3 = 20/3 = 6.6667- {4,4,13} = (4+4+13)/3 = 21/3 = 7- {4,8,13} = (4+8+13)/3 = 25/3 = 8.3333- {8,8,8} = (8+8+8)/3 = 24/3 = 8- {8,13,13} = (8+13+13)/3 = 34/3 = 11.3333Now, calculate the mean of these means: (6.6667 + 7 + 8.3333 + 8 + 11.3333)/5 = 41.3333/5 = 8.2667Then, calculate the standard deviation of these means:$$ \sqrt{\frac{[(6.6667-8.2667)^2 + (7-8.2667)^2 + (8.3333-8.2667)^2 + (8-8.2667)^2 + (11.3333-8.2667)^2]}{5}} $$$$ = \sqrt{\frac{(1.6^2 + 1.2667^2 + 0.0666^2 + 0.2667^2 + 3.0666^2)}{5}} $$$$ = \sqrt{\frac{(2.56 + 1.6044 + 0.0044 + 0.0711 + 9.4036)}{5}} $$$$ = \sqrt{\frac{13.6435}{5}} $$$$ = \sqrt{2.7287} $$$$ = 1.6518 $$It seems there was a mistake in my calculation. Let's correct the approach for calculating the standard deviation of the means:First, calculate the variance of the means:$$ \text{Variance} = \frac{[(6.6667-8.2667)^2 + (7-8.2667)^2 + (8.3333-8.2667)^2 + (8-8.2667)^2 + (11.3333-8.2667)^2]}{4} $$ (since for standard deviation of a sample, we divide by n-1, here n=5)$$ \text{Variance} = \frac{(-1.6)^2 + (-1.2667)^2 + (0.0666)^2 + (-0.2667)^2 + (3.0666)^2}{4} $$$$ \text{Variance} = \frac{2.56 + 1.6044 + 0.0044 + 0.0711 + 9.4036}{4} $$$$ \text{Variance} = \frac{13.6435}{4} $$$$ \text{Variance} = 3.410875 $$Then, the standard deviation (which is the square root of the variance):$$ \text{Standard Deviation} = \sqrt{3.410875} \approx 1.8469 $$Therefore, the correct answer is B) 1.8469, which is the bootstrap estimate of the standard error of the sample mean.

Consider the Following Sample of Three Measurements {4,8,13}