Use the relatively small number of given bootstrap samples to construct the
confidence interval.
Here is a sample of measured radiation emissions (cW/kg) for cell phones
(based on data from the Environmental Working Group): \(38,55,86,145 .\) Her
bootstrap samples:
\(\\{38,145,55,86\\},\\{86,38,145,145\\},\\{145,86,55,55\\},\\{55,55,55,145\\},$$\\{86,86,55,55\\},\\{38,38,86,86\\},\\{145,38,86,55\\},\\{55,86,86,86\\},\\{145,86,55,86\\}
\\{38,145,86,55\\}\).
a. Using only the ten given bootstrap samples, construct an \(80 \%\) confidence
interval estimate of the population mean.
b. Using only the ten given bootstrap samples, construct an \(80 \%\) confidence
interval estimate of the population standard deviation.
The 80% CI for the mean is [70.5, 93]. The 80% CI for the SD is [17.33, 47.55].
Step by step solution
01
- Calculate Means of Bootstrap Samples
First, identify the mean of each bootstrap sample. \[\{38,145,55,86\} \Rightarrow \text{mean} = \frac{38 + 145 + 55 + 86}{4} = 81 \]\[\{86,38,145,145\} \Rightarrow \text{mean} = \frac{86 + 38 + 145 + 145}{4} = 103.5 \]\[\{145,86,55,55\} \Rightarrow \text{mean} = \frac{145 + 86 + 55 + 55}{4} = 85.25 \]\[\{55,55,55,145\} \Rightarrow \text{mean} = \frac{55 + 55 + 55 + 145}{4} = 77.5 \]\[\{86,86,55,55\} \Rightarrow \text{mean} = \frac{86 + 86 + 55 + 55}{4} = 70.5 \]\[\{38,38,86,86\} \Rightarrow \text{mean} = \frac{38 + 38 + 86 + 86}{4} = 62 \]\[\{145,38,86,55\} \Rightarrow \text{mean} = \frac{145 + 38 + 86 + 55}{4} = 81 \]\[\{55,86,86,86\} \Rightarrow \text{mean} = \frac{55 + 86 + 86 + 86}{4} = 78.25 \]\[\{145,86,55,86\} \Rightarrow \text{mean} = \frac{145 + 86 + 55 + 86}{4} = 93 \]\[\{38,145,86,55\} \Rightarrow \text{mean} = \frac{38 + 145 + 86 + 55}{4} = 81 \]
02
- Order the Means and Find the 80% Confidence Interval
Order the sample means and choose the central 8 observations out of the ten. The ordered means are: 62, 70.5, 77.5, 78.25, 81, 81, 81, 85.25, 93, 103.5. For an 80% confidence interval, we exclude the 1st and the 10th observations: \[ \text{80\text(% ) Confidence Interval } = [70.5, 93] \].
03
- Calculate Standard Deviations of Bootstrap Samples
Identify the standard deviation of each bootstrap sample. \[\{38,145,55,86\} \Rightarrow \text{SD} \approx 47.55 \]\[\{86,38,145,145\} \Rightarrow \text{SD} \approx 54.59 \]\[\{145,86,55,55\} \Rightarrow \text{SD} \approx 42.68 \]\[\{55,55,55,145\} \Rightarrow \text{SD} \approx 41.94 \]\[\{86,86,55,55\} \Rightarrow \text{SD} \approx 17.33 \]\[\{38,38,86,86\} \Rightarrow \text{SD} \approx 24 \]\[\{145,38,86,55\} \Rightarrow \text{SD} \approx 41.94 \]\[\{55,86,86,86\} \Rightarrow \text{SD} \approx 14.28 \]\[\{145,86,55,86\} \Rightarrow \text{SD} \approx 35.77 \]\[\{38,145,86,55\} \Rightarrow \text{SD} \approx 47.03 \]
04
- Order the Standard Deviations and Find the 80% Confidence Interval
Order the standard deviations and choose the central 8 observations out of the ten. The ordered standard deviations are: 14.28, 17.33, 24, 35.77, 41.94, 41.94, 42.68, 47.03, 47.55, 54.59. For an 80% confidence interval, we exclude the 1st and the 10th observations: \[ \text{80\text(% ) Confidence Interval } = [17.33, 47.55] \].
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Key Concepts
These are the key concepts you need to understand to accurately answer the question.
confidence interval
A confidence interval provides the range within which we expect the true value (like a population mean) to fall. It鈥檚 based on sample data and gives us an estimate of the uncertainty surrounding our results. Imagine you are trying to estimate the average height of all students in a school. You can't measure everyone, so you measure a sample. The confidence interval tells us that we can be, say, 80% certain that the true average height falls within a specific range.
To construct a confidence interval, you need the sample mean, the variability of the data, and the desired confidence level (like 80%, 90%, or 95%). In our example, we use bootstrap samples to generate multiple sample means and then determine the range where most of these means fall. This range is our confidence interval.
bootstrap method
The bootstrap method is a powerful statistical technique used to estimate the properties of an estimator (like its variance) by sampling with replacement from the original data. This approach allows us to construct confidence intervals without making assumptions about the data distribution.
Think of it as creating many 'mini-samples' from your original data set by randomly selecting data points with replacement. By calculating the mean or standard deviation of these mini-samples, we can build a distribution of these statistics and use it to create confidence intervals.
- Easy to implement
- No assumptions about the data distribution
- Useful for small samples
The bootstrap method is particularly valuable when you have limited data and need a robust way to gauge the uncertainty in your estimates.
statistical analysis
Statistical analysis allows us to make sense of our data. It involves collecting, organizing, interpreting, and presenting data to discover underlying patterns or trends. This can include calculating measures of central tendency (like the mean or median), variability (such as variance or standard deviation), and more advanced techniques like hypothesis testing and regression analysis.
In the context of our exercise, statistical analysis involves calculating the means and standard deviations of bootstrap samples and then constructing confidence intervals. This helps us understand the range within which our population parameters are likely to fall.
It's important because it provides a framework for making decisions based on data and quantifies uncertainty, making our conclusions more reliable.
population mean
The population mean is the average value of a particular characteristic (like height, weight, or in our example, radiation emissions) for the entire population. In many real-world situations, measuring the entire population is not feasible, so we use sample data to estimate the population mean.
The sample mean is calculated by adding up all the sample data points and dividing by the number of points. By using bootstrap samples, we determine multiple sample means, which help us estimate the true population mean and provide the confidence interval.
- Calculates the central tendency
- Helps in making predictions about the population
- Aids in understanding the overall behavior of the population
Understanding the population mean is crucial in statistical analysis as it forms the basis for many other calculations and interpretations.
population standard deviation
The population standard deviation measures the amount of variability or dispersion in a population. It tells us how much the individual data points typically deviate from the population mean. A small standard deviation indicates that the data points are close to the mean, while a large standard deviation indicates more spread out data points.
Just like with the population mean, calculating the standard deviation for an entire population is often impractical. Instead, we use sample data to estimate it. In our exercise, we calculate the standard deviations of bootstrap samples and create a confidence interval for the population standard deviation.
Understanding the population standard deviation is important for:
- Identifying the spread of the data
- Assessing variability within the population
- Making informed decisions based on the consistency of the data
It provides insights into how consistent the data is, which is essential for many areas of statistical analysis.
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