Bootstrapping is sampling with replacement from a sample that is representative

Question

Bootstrapping is sampling with replacement from a sample that is representative of the population whose parameters we wish to estimate. You will obtain many random samples with replacement from the sample data and compute the mean of each random sample. For a 80% confidence interval, you will find the cutoff points for the middle 80% of the sample means. That is, you will find the 10th and 90th percentiles. These cutoff points represent the lower and upper bounds of the confidence interval. The following data represents the miles per gallon that 10 individuals experienced with their 2009 Smart car with a 1.0 Liter engine and c transmission 34.0 34.6 39.8 36.6 42.9 46.3 35.3 32.3 43.8 Source: fueleconomy gov 1) Create 1000 means from "bootstrap samples" of size 10. Follow the path Applets-Resampling-Bootstrap a Statistic Find the 10th and 90th percentiles. 2) Find the 80% T-interval for the population mean mpg for all 2009 Smart cars, using the data in the table above. Use StatCrunch and check with the TI calculator. What assumption has to be checked first before you can use the formulas and technology? 3) Compare the two intervals from steps 1 and 2. Are they similar? 4) In what cases is "bootstrapping" most useful? Do we need to use it in this case?

Explanation / Answer

Descriptive Statistics for miles/gallon along with bootstrap results based on 1000 bootstrap samples and confidence intervals at 80%

Descriptive Statistics

Statistic

Std. Error

Bootstrapa

Bias

Std. Error

80% Confidence Interval

Lower

Miles/gallon

N

10

0

0

10

10

Range

14.00

Minimum

32.30

Maximum

46.30

Mean

38.7100

1.51397

-.0549

1.4253

36.8110

40.5200

Std. Deviation

4.78759

-.29229

.72332

3.56021

5.43684

Variance

22.921

-2.191

6.351

12.675

29.559

Skewness

.227

.687

.001

.645

-.561

1.043

Kurtosis

-1.439

1.334

.470

1.191

-1.947

.354

Valid N (listwise)

N

10

0

0

10

10

a. Unless otherwise noted, bootstrap results are based on 1000 bootstrap samples

Below R code- generates 1000 bootstrap means along with calculation of 10th percentile and 90th percentile.

###############################################################

y<-c(34.0,34.6,39.8,36.6,42.9,41.5,46.3,35.3,32.3,43.8) ### miles/gallon data
require(boot) ### Install package boot in R
med<- function(x,i) mean(x[i])
b1 <- boot(y,med,1000)
b1
b1$t #### 1000 bootstrap means
quantile(b1$t,0.1) #### 10th percentile
quantile(b1$t,0.9) #### 90th percentile
plot(b1,col=2) ##### Plot of means, sampling distribution of means

The basic idea of bootstrapping is that inference about a population from sample data, (sample population), can be modelled by resampling the sample data and performing inference about a sample from resampled data, (resampled sample). As the population is unknown, the true error in a sample statistic against its population value is unknown. In bootstrap-resamples, the 'population' is in fact the sample, and this is known; hence the quality of inference of the 'true' sample from resampled data, (resampled sample), is measurable.

Descriptive Statistics

Statistic

Std. Error

Bootstrapa

Bias

Std. Error

80% Confidence Interval

Lower

Miles/gallon

N

10

0

0

10

10

Range

14.00

Minimum

32.30

Maximum

46.30

Mean

38.7100

1.51397

-.0549

1.4253

36.8110

40.5200

Std. Deviation

4.78759

-.29229

.72332

3.56021

5.43684

Variance

22.921

-2.191

6.351

12.675

29.559

Skewness

.227

.687

.001

.645

-.561

1.043

Kurtosis

-1.439

1.334

.470

1.191

-1.947

.354

Valid N (listwise)

N

10

0

0

10

10

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