Problem 119 - An Introduction to Mathematical Statistics and Its Applications

Question

Suppose the charge for repairing an automobile averages \(\$ 200\) with a standard deviation of \(\$ 16\). If a \(10 \%\) \(\operatorname{tax}\) is added to the charge and then a \(\$ 15\) flat fee for environmental impact, what is the standard deviation of the charge to the car owner?

Step-by-Step Solution

Answer

The standard deviation of the charge to the car owner is \$ 17.6.

Step 1: Identify the Variables

The charge for repairing an automobile averages \$200 with a standard deviation of \$16. Thus, we know the standard deviation of the current charge \$x\$ is \$16. In this case the tax is 10\% which can be represented as 0.1, and the flat fee for environmental impact is \$15.

Step 2: Formulate the Function

The new charge will be the old charge plus 10\% of the old charge and a fixed \$15 environmental fee. This will give us the function: \(y = 1.1x + 15\).

Step 3: Calculate the Standard Deviation

From the properties of variance, we know that the variance of a function of a random variable is the square of the multiplier times the original variance, and the standard deviation is the square root of the variance. Since the standard deviation of the original charge is \$16 and the multiplier is 1.1, the standard deviation of the new charge is: \( \sqrt{ (1.1)^2 \times (16)^2 } = 1.1 \times 16 = 17.6 \)

Key Concepts

Variance

Variance is a key concept in statistics that measures how much a set of numbers is spread out. It tells us how much the individual numbers in a data set differ from the mean (average) of the set. The larger the variance, the more spread out the numbers are.

• Variance is calculated as the average of the squared differences from the mean.
• It is represented mathematically as \( \sigma^2 \).

In our example, the variance of the initial repair cost helps us understand the spread of the charges around the average of $200. The original variance can be derived by squaring the standard deviation, \(16^2\).
When the cost is adjusted for tax and a fee, the variance changes based on the multiplier used in the function, affecting the new calculation.

Random Variables

Random variables are variables whose possible values are numerical outcomes of a random phenomenon. In our context, the cost of automobile repair can be viewed as a random variable. The price varies depending on numerous factors, capturing the inherent uncertainty in repair costs.

• A random variable can be discrete or continuous, depending on whether it takes on fixed or any value in an interval, respectively.
• In statistics, we often deal with transformations of random variables, such as adding taxes or fees.

The function \(y = 1.1x + 15\) describes how our random variable is transformed by applying a linear function. The original values (x) are transformed into new values (y), factoring in the additional cost elements. This real-life example provides insights into how random variables operate in statistical calculations.

Statistics

Statistics is a discipline concerned with collecting, analyzing, interpreting, presenting, and organizing data. It uses methods to understand and interpret data sets and predict future trends.

• Descriptive statistics summarize basic features of data, like mean and variance.
• Inferential statistics make predictions or inferences about a population based on a sample.

In our example of automobile repair charges, statistics provide a way to summarize and predict the variation in charges. The standard deviation gives us an intuitive gauge of the charge's reliability around the average expense, informing decision-makers about pricing stability.

Mathematical Functions

Mathematical functions describe relationships between quantities. In our scenario, a function is used to calculate the new charge for car repairs after tax and fees.

• A function \( f(x) \) maps input (\(x\)) to an output, often expressed algebraically.
• The function \( y = 1.1x + 15 \) specifies how each original charge (\(x\)) is transformed into a new charge (\(y\)), factoring in the multiplier for tax and the additive fee for environmental impact.

The function impacts not only the mean of charges but also their spread, influencing both variance and standard deviation. Functions are fundamental in mathematics as they describe dynamic relationships and transformations in statistics and other disciplines.