Question

Tickets. A delivery company's trucks occasionally get parking tickets, and based on past experience, the company plans that the trucks will average \(1.3\) tickets a month, with a standard deviation of \(0.7\) tickets. a) If they have 18 trucks, what are the mean and standard deviation of the total number of parking tickets the company will have to pay this month? b) What assumption did you make in answering?

Short answer

Mean is 23.4 tickets; standard deviation is approximately 2.97 tickets. Assumes independent ticketing for each truck.

Step-by-step solution

Define Mean for a Single Truck

For a single truck, the mean number of tickets is given as 1.3 tickets per month. This is the average number of tickets each truck is expected to collect.

Calculate Total Mean for All Trucks

Since there are 18 trucks, the mean number of tickets for all trucks combined is simply the single truck mean multiplied by the number of trucks: \[ \text{Mean total} = 18 \times 1.3 = 23.4 \] This means collectively, the trucks are expected to receive an average of 23.4 tickets this month.

Define Standard Deviation for a Single Truck

For a single truck, the standard deviation of the number of tickets is given as 0.7 tickets per month. This represents the variability in the number of tickets from the mean for a single truck.

Calculate Total Standard Deviation for All Trucks

When calculating the total standard deviation for independent events, we use the formula for the standard deviation of a sum: \[ \text{SD total} = \sqrt{N} \times \text{SD single} \] where \( N \) is the number of independent events (trucks). For 18 trucks, \[ \text{SD total} = \sqrt{18} \times 0.7 \approx 2.97 \] This is the standard deviation of the total number of tickets for all trucks.

Assumption Made

The key assumption here is that the number of tickets each truck gets is independent from the other trucks. We assume no truck's parking behavior affects another, and each has a similar chance of receiving a ticket, with outcomes following a normal distribution.

Key concepts

Mean Calculation

Mean calculation is at the heart of statistical analysis. It helps us find the average, or expected value, of a dataset. In this exercise, the mean number of parking tickets for one truck is 1.3 per month. This represents the typical number of tickets a single truck might collect.

To find the mean for all 18 trucks, we simply multiply the mean for one truck by 18. This gives us an expected total of 23.4 tickets for the month.

• Mean for one truck: 1.3 tickets.
• Total mean for 18 trucks: \( 18 \times 1.3 = 23.4 \) tickets.

This calculation is crucial as it sets the foundation for further statistical analysis, like finding the variance and standard deviation.

Standard Deviation

Standard deviation measures how much values in a dataset differ from the mean. In our context, it shows the spread of parking tickets around the average number of tickets per truck. For a single truck, the standard deviation is 0.7.

To calculate the standard deviation for all 18 trucks, because each truck is independent, we use the formula for the standard deviation of a sum.

• Square root of the number of trucks: \( \sqrt{18} \).
• Total standard deviation: \( \text{SD total} = \sqrt{18} \times 0.7 \approx 2.97 \).

This means there is a moderate spread in the total number of tickets they might receive, considering all factors as constant.

Independent Events

In probability and statistics, independent events are those where the occurrence of one event does not affect another. Here, each truck getting a ticket does not impact whether another truck gets one.

Understanding independence is crucial as it allows us to calculate probabilities and statistical measures more easily. For example, because the trucks are independent:

• We can apply the sum of mean calculations across all trucks.
• We can calculate total standard deviation using the square root rule.

This assumption simplifies our analysis and is fundamental for correctly applying the normal distribution model.

Statistical Assumptions

Statistical assumptions form the basis of our calculations and interpretations. In this scenario, we assumed that the ticketing process is essentially random and normally distributed.

These assumptions were necessary:

• Each truck operates independently; no external factors influence the number of tickets they receive collectively.
• The number of tickets follows a normal distribution, which allows us to use tools like mean and standard deviation effectively.

If these assumptions hold true, our statistical conclusions will be accurate. Deviations from these assumptions may require a reassessment of the model.