Question

Automobile Depreciation For a random sample of 20 automobile models, we record the value of the model as a new car and the value after the car has been purchased and driven 10 miles. \({ }^{47}\) The difference between these two values is a measure of the depreciation on the car just by driving it off the lot. Depreciation values from our sample of 20 automobile models can be found in the dataset CarDepreciation. (a) Find the mean and standard deviation of the Depreciation amounts in CarDepreciation. (b) Use StatKey or other technology to create a bootstrap distribution of the sample mean of depreciations. Describe the shape, center, and spread of this distribution. (c) Use the standard error obtained in your bootstrap distribution to find and interpret a \(95 \%\) confidence interval for the mean amount a new car depreciates by driving it off the lot.

Short answer

Part a: Mean and standard deviation are calculated using statistical formulas. Part b: Bootstrap distribution is created by resampling the data with replacement and noting the shape, center and spread of this distribution. Part c: The 95% confidence interval is found using the formula (sample mean ± (1.96 * bootstrap standard error)) which is then interpreted in the context of the problem.

Step-by-step solution

Compute for the Mean and Standard Deviation

To find the mean, calculate the sum of all depreciation amounts and divide it by the total number of observations. The standard deviation can be computed by finding the square root of the variance. Variance is the average of the squared differences from the Mean.

Create Bootstrap Distribution

A bootstrap distribution of the sample mean is created by resampling the data with replacement. This process is repeated many times, each time saving the sample mean. The collection of these resampled means are used to form the bootstrap distribution.

Describe The Shape, Center, And Spread Of Distribution

Note the shape of the bootstrap distribution. It might be symmetrical (normal), positively skewed, negatively skewed, or bimodal. Then find the center (mean or median) and spread or variability (standard deviation or inter-quartile range) of the distribution.

Compute for the Confidence Interval

A 95% confidence interval can be computed as the sample mean ± (1.96 * bootstrap standard error). The bootstrap standard error can be found as the standard deviation of the bootstrap distribution.

Interpret the Confidence Interval

Interpret the 95% confidence interval in the context of the problem. It indicates that we are 95% confident that the interval contains the true mean depreciation amount for new cars driven off the lot.