/*! 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 76 According to a Gallup poll, \(92... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 6: Problem 76

According to a Gallup poll, \(92 \%\) of Americans believe in God (Time, June 20,2011 ). Suppose that this result is true for the current population of adult Americans. What is the probability that the number of adult Americans in a sample of 500 who believe in God is a. exactly 445 b. at least 450 c. 440 to 470

Short Answer

Expert verified
The probability of having exactly 445 people who believe in God in the sample of 500 is calculated using the Binomial distribution formula. For probabilities of 450 or more and between 440 and 470, we use Normal approximation of Binomial distribution.

Step by step solution

01

Identify the variables

The sample size \(n\) is 500, the probability of success \(p\) is 0.92, probability of failure \(q=1-p=0.08\).
02

Apply Binomial Formula to solve a

The formula for binomial probability is \(P(X=k) = ^nC_k * p^{k}* q^{n-k}, P(X=445) = ^500C445 * 0.92^{445}* 0.08^{55}\)
03

Use Normal approximation to solve b

The sample size is large, so for at least 450, normal approximation of binomial probibility can be used. \(P(X \geq 450) = 1- P(X<449)\). The mean \(碌 = np\) and standard deviation \(蟽 = \sqrt{npq}\) . Find \(Z=(X-碌)/蟽\), for \(X=449\), \(Z= (449-碌)/蟽\)
04

Use Normal approximation to solve c

Find \(P(440 \leq X \leq 470)\) using the normal table or calculator, we use \(Z = (440.5-碌)/蟽\) and \(Z= (469.5-碌)/蟽\) and subtract the smaller Z-table value from the larger one

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.

Probability
Probability measures the likelihood that a particular event will happen. In the context of this exercise, we want to find the probability that a certain number of people believe in God.

To break it down:
  • Event: Whether each individual in a sample believes in God.
  • Outcome: In each trial (asking a person), there are only two outcomes 鈥� they either believe in God or they don't.
  • Success Probability (\( p \)): The probability that any one individual believes in God, given as 92%, or 0.92.
  • Failure Probability (\( q \)): The probability that an individual doesn't believe in God, calculated as \(1 - p = 0.08 \).
  • Sample Size (\( n \)): The number of people surveyed, which is 500.
By understanding these components, we can use them to analyze various probabilities using the binomial distribution and its approximations.
Normal Approximation
When you have a large sample size, the binomial distribution can be approximated by the normal distribution. This makes calculations simpler. Instead of computing the probability for each possible outcome, we use the continuity correction and the normal curve.

Here's how it works:
  • Central Limit Theorem: As the sample size \( n \) increases, the binomial distribution approaches a normal distribution.
  • Mean (\( \mu \)): Calculated as \( np \). For our problem, \( \mu = 500 \times 0.92 = 460 \).
  • Standard Deviation (\( \sigma \)): Calculated as \( \sqrt{npq} \). For our problem, \( \sigma = \sqrt{500 \times 0.92 \times 0.08} = \sqrt{36.8} \).
  • Continuity Correction: When converting a discrete binomial distribution to a continuous normal distribution, add or subtract 0.5 from the desired value. For example, calculating \( P(X \geq 450) \) involves \( X = 449.5 \).
  • Standard Score (Z-score): Compute \( Z = \frac{X - \mu}{\sigma} \) to find the probability from the Z-table.
This method transforms the problem into one involving the normal distribution, which is often much easier to handle for large samples.
Binomial Formula
The binomial formula helps us determine the probability of achieving exactly \( k \) successes in \( n \) trials. This formula is essential for problems with a clear yes/no outcome, such as our exercise.

The formula is:
  • \( P(X = k) = \binom{n}{k} p^k q^{n-k} \), where \( \binom{n}{k} \) is the number of ways to choose \( k \) successes from \( n \) trials, also known as "n choose k."
  • In our problem:
    • For \( X = 445 \), \( \binom{500}{445} \) calculates the number of ways to have 445 believers out of 500 people.
    • \( p^{445} \) accounts for the probability of 445 people believing in God.
    • \( q^{55} \) is the probability of the remaining 55 people not believing in God.
By plugging these into our equation, we get the probability of finding exactly 445 believers from a sample size of 500.
Statistics Problem Solving
Solving statistics problems involves several steps to reach a solution effectively. Let's walk through our exercise to understand this approach better.

Here鈥檚 how you can approach a statistics problem like this:
  • Understand the Question: Identify what probability you need to find. Are you looking for an exact number, a range, or at least a certain number of successes?
  • Identify the Distribution: Determine if the problem fits a binomial distribution or if a normal approximation is needed for a larger sample.
  • Choose the Right Formula: Use the binomial formula for exact probabilities and the normal approximation for ranges or cumulative probabilities.
  • Calculate Correctly: Carefully compute values like the mean, standard deviation, and Z-scores.
  • Interpret Results: Make sure to interpret your results in the context of the question. What does this probability mean practically?
Approaching statistics problems systematically helps in avoiding errors and ensures a comprehensive understanding of the solution.

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

At Jen and Perry Ice Cream Company, a machine fills 1-pound cartons of Top Flavor ice cream. The machine can be set to dispense, on average, any amount of ice cream into these cartons. However, the machine does not put exactly the same amount of ice cream into each carton; it varies from carton to carton. It is known that the amount of ice cream put into each such carton has a normal distribution with a standard deviation of 18 ounce. The quality control inspector wants to set the machine such that at least \(90 \%\) of the cartons have more than 16 ounces of ice cream. What should be the mean amount of ice cream put into these cartons by this machine?

The Bank of Connecticut issues Visa and MasterCard credit cards. It is estimated that the balances on all Visa credit cards issued by the Bank of Connecticut have a mean of \(\$ 845\) and a standard deviation of \(\$ 270 .\) Assume that the balances on all these Visa cards follow a normal distribution. a. What is the probability that a randomly selected Visa card issued by this bank has a balance between \(\$ 1000\) and \(\$ 1440\) ? b. What percentage of the Visa cards issued by this bank have a balance of \(\$ 730\) or more?

Determine the area under a normal distribution curve with \(\mu=55\) and \(\sigma=7\) a. to the right of \(x=58\) b. to the right of \(x=43\) \(c .\) to the left of \(x=68\) d. to the left of \(x=22\)

One of the cars sold by Walt's car dealership is a very popular subcompact car called Rhino. The final sale price of the basic model of this car varies from customer to customer depending on the negotiating skills and persistence of the customer. Assume that these sale prices of this car are normally distributed with a mean of \(\$ 19,800\) and a standard deviation of \(\$ 350\). a. Dolores paid \(\$ 19,445\) for her Rhino. What percentage of Walt's customers paid less than Dolores for a Rhino? b. Cuthbert paid \(\$ 20,300\) for a Rhino. What percentage of Walt's customers paid more than Cuthbert for a Rhino?

Let \(x\) be a continuous random variable that is normally distributed with a mean of 80 and a standard deviation of \(12 .\) Find the probability that \(x\) assumes a value a. greater than 69 b. less than 73 c. greater than 101 d. less than 87
See all solutions

Recommended explanations on Math Textbooks

Decision Maths

Read Explanation

Logic and Functions

Read Explanation

Statistics

Read Explanation

Calculus

Read Explanation

Geometry

Read Explanation

Pure Maths

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.