/*! 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 99 Suppose that \(50 \%\) of all yo... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 5: Problem 99

Suppose that \(50 \%\) of all young adults prefer McDonald's to Burger King when asked to state a preference. A group of 100 young adults were randomly selected and their preferences recorded. a. What is the probability that more than 60 preferred McDonald's? b. What is the probability that between 40 and 60 (inclusive) preferred McDonald's? c. What is the probability that between 40 and 60 (inclusive) preferred Burger King?

Short Answer

Expert verified
Answer: The probability that more than 60 young adults preferred McDonald's is approximately 1.16%, while the probabilities that between 40 and 60 (inclusive) young adults preferred McDonald's and Burger King are both approximately 96.48%.

Step by step solution

01

Identify the binomial distribution

The binomial distribution is used when there are two possible outcomes to an experiment, each with a known probability. In this case, we have the preference of McDonald's or Burger King, with the probability of preferring McDonald's being 50%. With a sample size of 100, we have a binomial distribution with n=100 and the probability of success (preferring McDonald's) being p=0.5.
02

Calculate the required probabilities

To calculate the probability of specific scenarios within this binomial distribution, we will use the cumulative binomial probability function. This can be expressed as: \(P(X \le k) = \sum_{i=0}^{k} \binom{n}{i} p^i(1-p)^{(n-i)}\) Where \(P(X \le k)\) is the cumulative probability, \(k\) is the number of successes, \(n\) is the number of trials, and \(p\) is the probability of success. a. More than 60 preferred McDonald's means we need to find the probability of k=61 to 100. Since the cumulative probability function gives us the probability of k or less, we can use 1 minus the cumulative probability of k=60 to get the probability of more than 60. \(P(X > 60) = 1 - P(X \le 60) = 1 - \sum_{i=0}^{60} \binom{100}{i} 0.5^i(1-0.5)^{(100-i)}\) b. For the probability that between 40 and 60 (inclusive) young adults preferred McDonald's, we need to find the cumulative probability of k=60 and subtract the cumulative probability of k=39. \(P(40 \le X \le 60) = P(X \le 60) - P(X \le 39) = \left[\sum_{i=0}^{60} \binom{100}{i} 0.5^i(1-0.5)^{(100-i)}\right] - \left[\sum_{i=0}^{39} \binom{100}{i} 0.5^i(1-0.5)^{(100-i)}\right]\) c. If the number of young adults preferring Burger King falls between 40 and 60 (inclusive), then the number of young adults preferring McDonald's falls between 60 and 40 (inclusive). Thus, this probability will be equal to the probability calculated in part (b).
03

Compute and present the probabilities

Using a calculator or statistical software to compute the probabilities, we find: a. \(P(X > 60) \approx 0.0116\) b. \(P(40 \le X \le 60) \approx 0.9648\) c. \(P(40 \le X \le 60) \approx 0.9648\) So the probability that more than 60 young adults preferred McDonald's is approximately 1.16%, while the probabilities that between 40 and 60 (inclusive) young adults preferred McDonald's and Burger King are both approximately 96.48%.

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.

Cumulative Binomial Probability
Understanding cumulative binomial probability is essential when facing problems like the one given, where we measure the likelihood of a certain number of successes in a sequence of yes/no experiments. In this example, the 'success' is a preference for McDonald's.

Cumulative binomial probability refers to the probability that the random variable takes on a value less than or equal to a specific number. Think of it as adding up the probabilities of all outcomes up to and including the outcome of interest. To visualize this, imagine stacking blocks up to a certain height鈥攃umulative probability is the total height of the stack, representing all the ways you can reach that height or lower.

To calculate it, we use the following formula:


\(P(X \le k) = \sum_{i=0}^{k} \binom{n}{i} p^i(1-p)^{(n-i)}\)

Here, \(n\) represents the number of trials, \(p\) the probability of success, \(k\) the number of successes, and \(\binom{n}{i}\) the binomial coefficient, which tells us the number of ways to choose \(i\) successes out of \(n\) trials. The sum of the probabilities from 0 to \(k\) gives us the cumulative probability.
Probability Calculation
Probability calculation is a fundamental process in statistics that helps us determine the likelihood of an event. With binomial probability problems, such as finding how many young adults prefer McDonald's, we are essentially dealing with calculations of 'what are the chances?'.

To calculate binomial probabilities, we employ the binomial probability formula which involves factorials:\[P(X = k) = \binom{n}{k} p^k(1-p)^{(n-k)}\]

The probability of more than 60 preferring McDonald's was found by calculating the probability of 60 or fewer and subtracting from one. For between 40 and 60, it involved finding the cumulative probability at both ends and taking the difference, which is a common method for calculating the probability of a range of outcomes.

Understanding each component of these calculations is key. The probability \(p\) must be known or estimated, \(n\) is the total number of trials, and \(k\) represents the number of successes. When these values are substituted into the formula, the calculations yield the probability of these specific events.
Random Sampling
Random sampling is the cornerstone of many statistical analyses, ensuring that the results are representative of the whole population. In the context of our problem, 100 young adults were 'randomly selected'. This randomness is crucial to avoid bias and gives every individual in the population an equal chance of being included in the sample.

When we say the sample is random, we mean each young adult had a 50% chance of being someone who prefers McDonald's (or Burger King). This principle allows us to use the binomial distribution model to predict preferences for a much larger population, assuming a stable preference rate.

It is important to note that random sampling is a method, not just a concept. It requires meticulous selection processes, often using random number generators or similar methods to ensure fairness and objectivity in the sample. It's a powerful tool that, when correctly applied, can give us a high level of confidence in our probability estimations and predictions.

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

Seeds are often treated with a fungicide for protection in poor-draining, wet environments. In a small-scale trial prior to a large-scale experiment to determine what dilution of the fungicide to apply, five treated seeds and five untreated seeds were planted in clay soil and the number of plants emerging from the treated and untreated seeds were recorded. Suppose the dilution was not effective and only four plants emerged. Let \(x\) represent the number of plants that emerged from treated seeds. a. Find the probability that \(x=4\). b. Find \(P(x \leq 3)\). c. Find \(P(2 \leq x \leq 3)\).

Insulin-dependent diabetes (IDD) is a common chronic disorder of children. This disease occurs most frequently in persons of northern European descent, but the incidence ranges from a low of \(1-2\) cases per 100,000 per year to a high of more than 40 per 100,000 in parts of Finland. \({ }^{11}\) Let us assume that an area in Europe has an incidence of 5 cases per 100,000 per year. a. Can the distribution of the number of cases of IDD in this area be approximated by a Poisson distribution? If so, what is the mean? b. What is the probability that the number of cases of IDD in this area is less than or equal to 3 per \(100,000 ?\) c. What is the probability that the number of cases is greater than or equal to 3 but less than or equal to 7 per \(100,000 ?\) d. Would you expect to observe 10 or more cases of IDD per 100,000 in this area in a given year? Why or why not?

Consider a Poisson random variable with \(\mu=2.5 .\) Calculate the following probabilities using the following table. $$ \begin{array}{|l|l|l|} \hline \text { Probability } & \text { Formula } & \text { Calculated Value } \\\ \hline P(x=0) & \frac{\mu^{k} e^{-\mu}}{k !}= & \\ \hline P(x=1) & \frac{\mu^{k} e^{-\mu}}{k !}= & \\ \hline P(x=2) & \frac{\mu^{k} e^{-\mu}}{k !}= & \\ \hline P(2 \text { or fewer successes }) & P(x=\longrightarrow)+P(x=\longrightarrow)+P(x=\longrightarrow) & \\ \hline \end{array} $$

Let \(x\) be a binomial random variable with \(n=\) 10 and \(p=.4 .\) Find these values: a. \(P(x=4)\) b. \(P(x \geq 4)\) c. \(P(x>4)\) d. \(P(x \leq 4)\) e. \(\mu=n p\) f. \(\sigma=\sqrt{n p q}\)

Let \(x\) be a hypergeometric random variable with \(N=15, n=3,\) and \(M=4\) a. Calculate \(p(0), p(1), p(2),\) and \(p(3)\) b. Construct the probability histogram for \(x\). c. Use the formulas given in Section 5.4 to calculate \(\mu=E(x)\) and \(\sigma^{2} .\) d. What proportion of the population of measurements fall into the interval \((\mu \pm 2 \sigma) ?\) Into the interval \((\mu \pm 3 \sigma)\) ? Do these results agree with those given by Tchebysheff's Theorem?
See all solutions

Recommended explanations on Math Textbooks

Statistics

Read Explanation

Discrete Mathematics

Read Explanation

Pure Maths

Read Explanation

Geometry

Read Explanation

Applied Mathematics

Read Explanation

Logic and Functions

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.