/*! This file is auto-generated */ .wp-block-button__link{color:#fff;background-color:#32373c;border-radius:9999px;box-shadow:none;text-decoration:none;padding:calc(.667em + 2px) calc(1.333em + 2px);font-size:1.125em}.wp-block-file__button{background:#32373c;color:#fff;text-decoration:none} Problem 3 Sample for a Poll There are 15,5... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 4: Problem 3

Sample for a Poll There are 15,524,971 adults in Florida. If The Gallup organization randomly selects 1068 adults without replacement, are the selections independent or dependent? If the selections are dependent, can they be treated as being independent for the purposes of calculations?

Short Answer

Expert verified
The selections are dependent but can be treated as independent due to the large population size.

Step by step solution

01

Understand the Definition of Independent and Dependent Events

Two events are independent if the occurrence of one does not affect the probability of the occurrence of the other. If the occurrence of one event affects the probability of the occurrence of the other, the events are dependent.
02

Determine the Nature of the Selection

Since adults are being selected without replacement, once an individual is selected, they cannot be chosen again. This means that the total number of adults changes with each selection, affecting the probability of selecting another adult. Therefore, the selections are dependent.
03

Evaluate the Impact of Dependence

Even though the selections are technically dependent, if the population is large enough, the effect of one selection on the probability of future selections is minimal. In this case, 1068 is a very small fraction of 15,524,971. The change in probability is so small that it can be ignored for practical purposes.
04

Conclusion

Although the selections are technically dependent, the vast size of the population compared to the sample size allows us to treat the selections as independent for the purposes of calculations.

Unlock Step-by-Step Solutions & Ace Your Exams!

  • Full Textbook Solutions

    Get detailed explanations and key concepts

  • Unlimited Al creation

    Al flashcards, explanations, exams and more...

  • Ads-free access

    To over 500 millions flashcards

  • Money-back guarantee

    We refund you if you fail your exam.

Over 30 million students worldwide already upgrade their learning with 91影视!

Key Concepts

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

independent events
In statistics, events are considered independent if the occurrence of one event does not affect the probability of the other event happening. Imagine flipping a coin twice. The outcome of the first flip (heads or tails) does not affect the outcome of the second flip. Each flip is an independent event because the result of one has no influence on the other. This property makes calculations straightforward since each event stands alone.
dependent events
Dependent events are the opposite of independent events. When the occurrence of one event affects the probability of another event, they are dependent. For example, consider drawing cards from a deck. If you draw a card and do not replace it, the probabilities change for the next draw. The probability of drawing an Ace on your second draw is affected by whether an Ace was drawn first. This dependency complicates calculations because the events are linked.
probability
Probability measures the likelihood of an event happening. It is calculated as the number of favorable outcomes divided by the total number of possible outcomes. For simple independent events like rolling a die, where there are six possible outcomes (1-6), the probability of rolling a 3 is 1/6. For dependent events, such as drawing cards without replacement, the probabilities change as the events unfold. Probability underpins all statistical studies, including the sampling method in the given exercise.
large population approximation
When dealing with very large populations, even if sampling is done without replacement, the selected samples can be treated as independent for practical calculations. This is because the change in probability after each selection becomes negligible. For example, selecting 1068 adults from a population of 15,524,971 is such a small portion that the effect on subsequent probabilities is minimal. This approximation simplifies calculations and is very useful in statistical surveys and polls.

One App. One Place for Learning.

All the tools & learning materials you need for study success - in one app.

Get started for free

Most popular questions from this chapter

Find the probability.At Least One. In Exercises \(5-12,\) find the probability. Find the probability that when a couple has three children, at least one of them is a girl. (Assume that boys and girls are equally likely.)

Express all probabilities as fractions. Current rules for telephone area codes allow the use of digits \(2-9\) for the first digit, and \(0-9\) for the second and third digits. How many different area codes are possible with these rules? That same rule applies to the exchange numbers, which are the three digits immediately preceding the last four digits of a phone number. Given both of those rules, how many 10-digit phone numbers are possible? Given that these rules apply to the United States and Canada and a few islands, are there enough possible phone numbers? (Assume that the combined population is about \(400,000,000 .\) )

Express all probabilities as fractions. As of this writing, the Mega Millions lottery is run in 44 states. Winning the jackpot requires that you select the correct five different numbers between 1 and 75 and, in a separate drawing, you must also select the correct single number between 1 and \(15 .\) Find the probability of winning the jackpot. How does the result compare to the probability of being struck by lightning in a year, which the National Weather Service estimates to be \(1 / 960,000 ?\)

Probability of a Run of Three Use a simulation approach to find the probability that when five consecutive babies are born, there is a run of at least three babies of the same sex. Describe the simulation procedure used, and determine whether such runs are unlikely.

Simulating Dice Assume that you have access to a computer that can randomly generate whole numbers between any two values. Describe how this computer can be used to simulate the rolling of a pair of dice.
See all solutions

Recommended explanations on Math Textbooks

Discrete Mathematics

Read Explanation

Applied Mathematics

Read Explanation

Geometry

Read Explanation

Mechanics Maths

Read Explanation

Decision Maths

Read Explanation

Statistics

Read Explanation
View all explanations

What do you think about this solution?

We value your feedback to improve our textbook solutions.

Study anywhere. Anytime. Across all devices.