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Chapter 1: Problem 14

Use a random-number table to simulate the outcomes of tossing a quarter 25 times. Assume that the quarter is balanced (i.e., fair).

Short Answer

Expert verified
Simulate 25 coin tosses using a random-number table, assign 0-4 as heads and 5-9 as tails, count the outcomes.

Step by step solution

01

Understand the Problem

We need to simulate 25 coin tosses using a random-number table. Each toss of a fair coin has two possible outcomes: heads, with probability 0.5, or tails, with probability 0.5.
02

Assign Numbers to Outcomes

To simulate the coin toss, we associate each number with an outcome: 0-4 to heads, and 5-9 to tails. This division ensures each outcome is equally likely.
03

Use Random-Number Table

Using the random-number table, select 25 digits. Each digit, from 0 to 9, will represent a simulated coin toss result according to the assignment in Step 2.
04

Record the Results

Go through the random-number table, pick 25 digits, and record the result as 'H' for heads if the digit is between 0 and 4, and 'T' for tails if the digit is between 5 and 9.
05

Count the Outcomes

After recording each result as heads or tails, count the total number of 'H's and 'T's for the entire simulation.

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Key Concepts

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

Random-Number Table
A random-number table is a tool used in probability and statistics that helps simulate random events or generate random numbers. It's essentially a list of numbers with no discernible pattern. This randomness makes it an excellent resource for simulations where fair chances need to be represented.
  • The numbers in the table range typically from 0 to 9.
  • Each digit in a random-number table represents an equally likely outcome, making it useful for simulators trying to model simple events such as coin tosses or dice rolls.
  • The lack of predictability in number sequences in a random-number table ensures fairness in simulations.
To utilize a random-number table in simulations, you first need to assign certain digits to particular outcomes, just as in the exercise with heads and tails. This approach enables easy visualization of how often each outcome appears compared to a completely random process.
Coin Toss Simulation
Simulating a coin toss using a random-number table can reflect the random nature of a real coin toss. In a fair coin toss, there's an equal chance of landing a heads or tails, and our simulation aims to mimic this. Here's how the simulation works:
  • Assign a segment of digits (0-4) to represent heads and another (5-9) to represent tails.
  • This ensures an equal probability of occurrence for each outcome in a simulated environment.
  • By using the random-number table, select digits at random and map them to heads or tails according to your initial assignment.
This simulation method helps learn about principles of randomness and probability without needing an actual coin. It allows students to perform numerous virtual trials efficiently and observe how random processes behave over multiple iterations.
Fair Coin Outcomes
In probability terms, a fair coin means that the chances of getting heads or tails are equal, typically viewed as a balanced coin.
  • The probability for each side (heads or tails) is 0.5.
  • Based on these probabilities, a fair simulation should result in approximately 50% heads and 50% tails over a large number of trials.
  • However, in a smaller number of tosses, such as 25, slight variations are natural due to the randomness inherent in each toss.
In an ideal simulation using a random-number table, distributing digits evenly amongst the outcomes (0-4 for heads, 5-9 for tails) mimics this fair probability distribution accurately. Any deviation in the counts of heads or tails provides a practical lesson in the variability inherent in random processes.

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Most popular questions from this chapter

For a set population, does a parameter ever change? If there are three different samples of the same size from a set population, is it possible to get three different values for the same statistic?

Modern Managed Hospitals (MMH) is a national for-profit chain of hospitals. Management wants to survey patients discharged this past year to obtain patient satisfaction profiles. They wish to use a sample of such patients. Several sampling techniques are described below. Categorize each technique as simple random sample, stratified sample, systematic sample, cluster sample, or convenience sample. (a) Obtain a list of patients discharged from all MMH facilities. Divide the patients according to length of hospital stay ( 2 days or less, \(3-7\) days, \(8-14\) days, more than 14 days). Draw simple random samples from each group. (b) Obtain lists of patients discharged from all MMH facilities. Number these patients, and then use a random-number table to obtain the sample. (c) Randomly select some MMH facilities from each of five geographic regions, and then include all the patients on the discharge lists of the selected hospitals. (d) At the beginning of the year, instruct each MMH facility to survey every 500th patient discharged. (e) Instruct each MMH facility to survey 10 discharged patients this week and send in the results.

Consider the students in your statistics class as the population and suppose they are seated in four rows of 10 students each. To select a sample, you toss a coin. If it comes up heads, you use the 20 students sitting in the first two rows as your sample. If it comes up tails, you use the 20 students sitting in the last two rows as your sample. (a) Does every student have an equal chance of being selected for the sample? Explain. (b) Is it possible to include students sitting in row 3 with students sitting in row 2 in your sample? Is your sample a simple random sample? Explain. (c) Describe a process you could use to get a simple random sample of size 20 from a class of size \(40 .\)

General: Gathering Data Which technique for gathering data (sampling, experiment, simulation, or census) do you think was used in the following studies? (a) An analysis of a sample of 31,000 patients from New York hospitals suggests that the poor and the elderly sue for malpractice at one-fifth the rate of wealthier patients (Journal of the American Medical Association). (b) The effects of wind shear on airplanes during both landing and takeoff were studied by using complex computer programs that mimic actual flight. (c) A study of all league football scores attained through touchdowns and field goals was conducted by the National Football League to determine whether field goals account for more scoring events than touchdowns (USA Today). (d) An Australian study included 588 men and women who already had some precancerous skin lesions. Half got a skin cream containing a sunscreen with a sun protection factor of \(17 ;\) half got an inactive cream. After 7 months, those using the sunscreen with the sun protection had fewer new precancerous skin lesions (New England Journal of Medicine).

A study of college graduates involves three variables: income level, job satisfaction, and one-way commute times to work. List some ways the variables might be confounded.
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