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Chapter 16: Problem 33

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

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

Step by step solution

01

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.
02

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.
03

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.
04

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.
05

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.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

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.

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Most popular questions from this chapter

Kids. A couple plans to have children until they get a girl, but they agree that they will not have more than three children even if all are boys. (Assume boys and girls are equally likely.) a) Create a probability model for the number of children they might have. b) Find the expected number of children. c) Find the expected number of boys they'll have.

Casino. A casino knows that people play the slot machines in hopes of hitting the jackpot but that most of them lose their dollar. Suppose a certain machine pays out an average of \(\$ 0.92\), with a standard deviation of \(\$ 120 .\) a) Why is the standard deviation so large? b) If you play 5 times, what are the mean and standard deviation of the casino's profit? c) If gamblers play this machine 1000 times in a day, what are the mean and standard deviation of the casino's profit? d) Is the casino likely to be profitable? Explain.

Insurance. An insurance policy costs \(\$ 100\) and will pay policyholders \(\$ 10,000\) if they suffer a major injury (resulting in hospitalization) or \(\$ 3000\) if they suffer a minor injury (resulting in lost time from work). The company estimates that each year 1 in every 2000 policyholders may have a major injury, and 1 in 500 a minor injury only. a) Create a probability model for the profit on a policy. b) What's the company's expected profit on this policy? c) What's the standard deviation?

Carnival. A carnival game offers a \(\$ 100\) cash prize for anyone who can break a balloon by throwing a dart at it. It costs \(\$ 5\) to play, and you're willing to spend up to \(\$ 20\) trying to win. You estimate that you have about a \(10 \%\) chance of hitting the balloon on any throw. a) Create a probability model for this carnival game. b) Find the expected number of darts you'll throw. c) Find your expected winnings.

Cancelled flights. Mary is deciding whether to book the cheaper flight home from college after her final exams, but she's unsure when her last exam will be. She thinks there is only a \(20 \%\) chance that the exam will be scheduled after the last day she can get a seat on the cheaper flight. If it is and she has to cancel the flight, she will lose \(\$ 150\). If she can take the cheaper flight, she will save \(\$ 100\). a) If she books the cheaper flight, what can she expect to gain, on average? b) What is the standard deviation?
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