Consider the data below:
33.1068240, -6.3331469, 22.2903011, -6.9142991, -21.7899524, -12.1741625, 29.2973275, -34.2364631, -4.4113930, -25.7845033, -12.0375226, 17.9397860, -6.2030667, 0.3504070 , -19.4854146, -10.5979710, -27.1274107, 20.4247578, 17.0975034, -6.8416016, 0.8700698, -9.9103781, 8.3101956, -6.9263659, -23.7861226, -1.8070511, 8.9497832, 27.5315404, 5.9680535, -2.4057959, 30.4888656, 8.3810280, -12.4548588, -19.5759143
a. Write an R-script to estimate the sampling distribution of the mean for this data. Use 2000 bootstrap iterations. Upload a pic of your bootstrap histogram.
b. What is the bootstrap estimate of the mean and standard error?
c. What is the 95% bootstrap CI for the mean?
d. What is your R code for the 95% bootstrap CI for the mean?
To estimate the sampling distribution of the mean and obtain the bootstrap estimates, you can follow these steps using R programming language:
a. Write an R-script to estimate the sampling distribution of the mean for this data using 2000 bootstrap iterations.
Step 1: Save the given data in a vector.
```
data <- c(33.1068240, -6.3331469, 22.2903011, -6.9142991, -21.7899524, -12.1741625, 29.2973275, -34.2364631, -4.4113930, -25.7845033, -12.0375226, 17.9397860, -6.2030667, 0.3504070 , -19.4854146, -10.5979710, -27.1274107, 20.4247578, 17.0975034, -6.8416016, 0.8700698, -9.9103781, 8.3101956, -6.9263659, -23.7861226, -1.8070511, 8.9497832, 27.5315404, 5.9680535, -2.4057959, 30.4888656, 8.3810280, -12.4548588, -19.5759143)
```
Step 2: Use the `boot` function to perform bootstrap resampling and estimation.
```
library(boot)
# Define a function to calculate the mean
calculate_mean <- function(data, index) {
mean(data[index])
}
# Perform bootstrap resampling with 2000 iterations
boot_result <- boot(data, calculate_mean, R = 2000)
```
Step 3: Plot the bootstrap distribution using a histogram.
```
hist(boot_result$t, main = "Bootstrap Distribution of the Mean", xlab = "Mean", col = "lightblue")
```
Save the plotted histogram as an image (e.g., using the `png` or `jpeg` function).
b. To obtain the bootstrap estimate of the mean and standard error:
```
bootstrap_mean <- mean(boot_result$t)
bootstrap_se <- sd(boot_result$t)
```
c. To calculate the 95% bootstrap confidence interval for the mean, use the `boot.ci` function:
```
bootstrap_ci <- boot.ci(boot_result, type = "bca", conf = 0.95)
```
The `type = "bca"` parameter specifies the bias-corrected and accelerated (BCa) method, which is a common method for calculating bootstrap confidence intervals.
d. Here is the R code for calculating the 95% bootstrap confidence interval for the mean:
```
boot.ci(boot_result, type = "bca", conf = 0.95)
```
Remember to save the plotted histogram and print the estimated mean and standard error, as well as the bootstrap confidence interval, for further analysis or reporting.