Problem 84 - An Introduction to Mathematical Statistics and Its Applications

Question

A city has 74,806 registered automobiles. Each is required to display a bumper decal showing that the owner paid an annual wheel tax of \(\$ 50\). By law, new decals need to be purchased during the month of the owner's birthday. How much wheel tax revenue can the city expect to receive in November?

Step-by-Step Solution

Answer

The city can expect to collect approximately $312,525 in car tax revenue in November.

Step 1: Identify the given quantities

We're given that there are 74,806 registered cars in the city and that each owner is required to pay a wheel tax of $50 each year on their birthday.

Step 2: Determine the revenue collected each month

Assuming birthdays are uniformly distributed throughout the year, we'd expect, on average, 1/12 of car owners to have their birthday in any given month and thus pay their car tax in that month.

Step 3: Calculate the total revenue for November

To find the total expected revenue for November, we need to multiply the number of car owners expected to pay in November by the amount of the tax. The number of car owners expected to pay in November is (1/12) * 74806, and the tax is $50.

Step 4: Simplify to get the final answer

Simplify the above calculation to get the final answer. The total expected revenue for November is: \((1/12) * 74806 * $50\)

Key Concepts

Uniform Distribution

The concept of uniform distribution is essential in understanding how certain events are spread over a given period. Imagine dividing a year into 12 equal parts, one for each month. If every month has the same number of events, such as birthdays or deadline payments, this even spread represents a uniform distribution. In the context of the problem, it is assumed that birthdays—and therefore tax payments—are uniformly distributed across the months.

This means that about 1/12 of the total number of car owners are expected to fulfill their tax obligations each month. For November, this would imply calculating the revenue as if 1/12 of all car owners were born in that month. Evenly spreading events in this manner allows for simple predictions and planning. While reality might slight vary, uniform distribution serves as an effective statistical tool for making general forecasts in cases where data is evenly distributed.

Tax Revenue Calculation

Determining tax revenue in situations involving uniform distribution can be methodical yet straightforward. The revenue in question here depends on the number of car owners who will need to pay their annual tax for the month of November. First, calculate the expected number of cars paying the tax in November by using the total number of registered cars and applying the uniform distribution rule, so \[ \frac{74,806}{12} \]. Once this number is known, multiply it by the annual wheel tax amount of $50.

Thus, the calculation for expected revenue becomes \[ \left( \frac{1}{12} \times 74,806 \right) \times 50 \]. By applying this formula, the city can easily estimate its revenue without having to go through every individual record. Simplifying these numbers provides a cleaner estimate that is practical for financial and resource planning. When approached systematically, tax revenue calculations can significantly aid in anticipating and managing a city's financial resources.

Probability Expectations

Probability expectations help in forecasting how likely events are to occur within given time frames. In the problem, the expectation that birthdays are uniformly distributed is the basis for estimating monthly tax revenue. The idea is that, although exact individual data isn't used, statistical methods allow for predictions based on assumed probabilities.

For this problem, we assume each car owner's birthday—and therefore tax payment—is equally likely to fall in any month, leading to a straightforward prediction. This assumption is based on probability expectations, which use historical or assumed patterns to project future outcomes. When combined with uniform distribution, it creates a powerful tool for anticipating monthly occurrences without extensive data analysis. Such calculations are vital in resource management and can be applied to various domains where events occur at random yet are evenly distributed over time.