Consider the following sample of four measurements {2,5,11,14}.Five bootstrap samples of this sample produced the following resamples: {2,5,5,14},{5,11,11,11},{2,11,14,14}, $\{ 2,2,14,14 \}$ , $\{ 5,5,11,14 \}$ .Based on these five resamples,find the bootstrap of the standard error of the sample median.
A)8.9
B)2.9240
C)2.3333
D)8.55
E)1.3077

Question

Quizplus · Accepted Answer

To find the bootstrap standard error of the sample median, we need to first calculate the median of each bootstrap sample. The medians of the five resamples are as follows:

{2,5,5,14} -> median = 5

{5,11,11,11} -> median = 11

{2,11,14,14} -> median = 12.5

{2,2,14,14} -> median = 8

{5,5,11,14} -> median = 8

The sample median is the median of the original sample, which is (2+5+11+14)/4 = 8.

The bootstrap standard error can be calculated as the standard deviation of the bootstrapped medians. Using the formula for the sample standard deviation of a set of values:

s = sqrt(sum[(x_i - mean)^2]/(n-1))

where x_i are the values, mean is the sample mean, and n is the sample size.

We can calculate the standard deviation of the bootstrapped medians as follows:

s = sqrt(((5-8)^2 + (11-8)^2 + (12.5-8)^2 + (8-8)^2 + (8-8)^2)/4)

= sqrt((9 + 9.5^2 + 17.5^2)/4)

= sqrt(106.25)/2

= 2.9240

Therefore, the bootstrap standard error of the sample median is 2.9240, which corresponds to choice (B).

Consider the Following Sample of Four Measurements {2,5,11,14} {2,2,14,14}\{ 2,2,14,14 \}{2,2,14,14}

Consider the Following Sample of Four Measurements {2,5,11,14} $\{ 2,2,14,14 \}$