91Ó°ÊÓ

Is this valid? Determine whether each of the following simulation designs is valid. Justify your answer. (a) According to a recent survey, \(50 \%\) of people aged 13 and older in the United States are addicted to texting. To simulate choosing a random sample of 20 people in this population and seeing how many of them are addicted to texting, use a deck of cards. Shuffle the deck well, and then draw one card at a time. A red card means that person is addicted to texting; a black card means he isn't. Continue until you have drawn 20 cards (without replacement) for the sample. (b) A tennis player gets \(95 \%\) of his serves in play during practice (that is, the ball doesn't go out of bounds). To simulate the player hitting 5 serves, look at 5 pairs of digits going across a row in Table \(D .\) If the number is between 00 and 94 , the serve is in; numbers between 95 and 99 indicate that the serve is out.

Step by step solution

01

Analyze Simulation Design (a)

This simulation involves choosing a random sample of 20 people using a deck of cards. In this setup, red cards (26 in a deck) represent being addicted to texting, while black cards (also 26 in a deck) represent not being addicted. Each draw of a card is without replacement.

02

Validity Check for Design (a)

To accurately represent 50% addiction rate, half of the people in the sample should be addicted, similar to a deck having 50% red cards and 50% black cards. Since a deck has exactly 26 red and 26 black cards, using a deck of cards properly simulates the population characteristics. Consequently, this design is valid.

03

Analyze Simulation Design (b)

In this design, the player hitting 5 serves is simulated using Table D, where digits from 00 to 94 indicate a serve in, and 95 to 99 indicate a serve out, aligning precisely with the 95% success rate.

04

Validity Check for Design (b)

This method uses 100 possible outcomes (00-99) where 95% (00-94) means a successful serve and the rest are failures. Thus, this distribution matches the player's performance being 95%, and the design is considered valid.

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Key Concepts

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

Probability Simulation

Probability simulation is a technique used to mimic a process with random elements by using a model that represents the real-world system.
One major goal is to understand the probability of certain outcomes under defined conditions.
These simulations allow us to experiment and predict behaviors without real-world consequences. In exercises involving probability simulation, like determining addiction rates or serve success, it's important to accurately model the situation:

• Identify variables (e.g., percentage of addiction or successful serves)
• Use appropriate tools (e.g., cards, tables of digits)
• Simulate multiple trials to observe variability and potential outcomes

This helps in understanding how likely certain events are to occur. Remember, a good probability simulation often includes replacing or not replacing items (like cards) to match real-world scenarios.

Sampling Methods

Sampling methods are techniques used to select a subset of individuals from a population to predict or make inferences about the population.
These methods must ensure that the sample is representative of the entire population.
Effective sampling requires a clear strategy to minimize bias and capture diverse characteristics of the larger group. In the context of simulation exercises, sampling can involve:

• Random sampling (mixing a deck of cards before picking)
• Stratified sampling (ensuring different population segments are proportionally represented)
• Systematic sampling (using a fixed interval, like every 5th card)

Choosing the right method is crucial for ensuring that the sample reflects the probability distribution you're simulating.

Statistical Validity

Statistical validity refers to how well a study or simulation measures what it purports to measure.
It is crucial for ensuring the reliability and accuracy of the results generated from simulations or statistical models.
Validation requires reviewing if the simulation setup correctly represents real-world probabilities and conditions. Several aspects to consider include:

• Correct use of randomness (e.g., shuffling cards thoroughly)
• Matching theoretical probability with experimental conditions with precise representations
• Checking that the outcomes align with expected probabilities

In the example scenarios, both simulations adhere to these validity conditions by accurately modeling addiction and serve probabilities.

Random Sampling

Random sampling is a foundational element in statistics that involves selecting a sample in such a way that each member of the population has an equal chance of being chosen.
It prevents bias and ensures that the sample represents the larger population accurately. In practical simulations, random sampling is often used to model scenarios:

• Simulating addiction with a deck of cards ensures each person's status is independent
• Digit tables use random numbers to assign outcomes, modeling randomness in actions like serving

By applying random sampling techniques, the simulation becomes a powerful tool for predicting real-world phenomena with minimal bias.