Asked by Clash Clans on Jun 10, 2024
Verified
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}\{ 2,2,14,14 \}{2,2,14,14} , {5,5,11,14}\{ 5,5,11,14 \}{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
Standard Error
A measure of the variability or dispersion of a sample statistic, indicating its precision as an estimate of the population parameter.
Sample Median
The middle value of a sample when the values are arranged in ascending or descending order, or the average of the two middle values if there is an even number of observations.
Resamples
The process of taking repeated samples from observed data, often used for bootstrapping or validating statistical models.
- Master the calculation and understanding of the standard error in assorted sample statistics via the bootstrap approach.
Verified Answer
{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).
Learning Objectives
- Master the calculation and understanding of the standard error in assorted sample statistics via the bootstrap approach.
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