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Chapter 5: Problem 27

Find the probabilities and answer the questions. Based on a Harris Interactive poll, \(20 \%\) of adults believe in reincarnation. Assume that six adults are randomly selected, and find the indicated probability. a. What is the probability that exactly five of the selected adults believe in reincarnation? b. What is the probability that all of the selected adults believe in reincarnation? c. What is the probability that at least five of the selected adults believe in reincarnation? d. If six adults are randomly selected, is five a significantly high number who believe in reincarnation?

Short Answer

Expert verified
a. P(X=5) 鈮� 0.00154b. P(X=6) 鈮� 0.000064c. P(X鈮�5) 鈮� 0.001604d. Yes, it is significantly high

Step by step solution

01

Understanding the Problem

We are given that 20\text{ \textbackslash \textpercent} of adults believe in reincarnation. We need to find probabilities involving six adults. This can be solved using the binomial probability formula: \[ P(X=x) = \binom{n}{x} \times p^x \times (1-p)^{n-x} \] where: - \( n = 6 \) (number of trials), - \( p = 0.2 \) (probability of success), and - \( x \) is the number of successes.
02

Find the Probability of Exactly Five Adults Believing in Reincarnation

Using the binomial probability formula: \[ P(X=5) = \binom{6}{5} \times 0.2^5 \times 0.8^1 \]\[ \binom{6}{5} = 6, \text{thus}, \]\[ P(X=5) = 6 \times (0.2)^5 \times (0.8) \]\[ P(X=5) 鈮� 0.00154 \]
03

Find the Probability of All Adults Believing in Reincarnation

Using the binomial probability formula: \[ P(X=6) = \binom{6}{6} \times 0.2^6 \times 0.8^0 \]\[ \binom{6}{6} = 1, \text{thus}, \]\[ P(X=6) = 1 \times (0.2)^6 \times (1) \]\[ P(X=6) 鈮� 0.000064 \]
04

Find the Probability of At Least Five Adults Believing in Reincarnation

We need to find the combined probability of exactly five and exactly six adults believing in reincarnation.\[ P(X \textgreater \text{or equal to} 5) = P(X=5) + P(X=6) \]\[ P(X \textgreater \text{or equal to} 5) = 0.00154 + 0.000064 \]\[ P(X \textgreater \text{or equal to} 5) 鈮� 0.001604 \]
05

Determine if Five Adults is a Significantly High Number

To determine if five adults is significantly high, compare the result from Step 1 with a significance level (commonly 0.05).Since \( P(X=5) \approx 0.00154 \) which is much less than 0.05, five adults believing in reincarnation is indeed a significantly high number.

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Key Concepts

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

Probability Distribution
Probability distribution gives us a way to handle and visualize probabilities of different outcomes. For this exercise, we're using a **binomial distribution**, which is a type of probability distribution. It works well when you have two possible outcomes - like 'success' and 'failure'. Here, 'success' is defined as an adult believing in reincarnation. Given that 20% of adults believe in reincarnation, we need to find probabilities for different counts of believers among six adults. This involves calculating the likelihood of specific outcomes and is foundational for solving these types of problems.
Binomial Formula
The binomial formula helps us find the probability of having exactly **x** successes in **n** trials. It looks like this: \( P(X=x) = \binom{n}{x} \times p^x \times (1-p)^{n-x} \) Breaking it down:
  • \( n \) is the number of trials (here, 6 adults)
  • \( x \) is the number of successful outcomes (e.g., adults who believe in reincarnation)
  • \( p \) is the probability of success for each trial (0.2 or 20% in this case)
If we want to find the probability that exactly five adults believe in reincarnation, set \( n = 6 \), \( x = 5 \), and \( p = 0.2 \): Using the formula, \( P(X=5) = \binom{6}{5} \times (0.2)^5 \times (0.8)^1 \). After calculating, \( P(X=5) \approx 0.00154 \).
Significance Level
The significance level helps us determine if an event is notable. It's often set at 0.05 (5%), meaning we consider results with a probability lower than this as statistically significant. In our exercise, we evaluated if it's significantly high for five adults among six to believe in reincarnation. We computed the probability as \( P(X=5)\approx 0.00154 \). Since this is much lower than 0.05, we conclude it's a significantly high number. In other words, it is quite uncommon for five out of six adults to believe in reincarnation, given the starting probability.
Statistical Analysis
Statistical analysis lets us make informed conclusions based on data. By applying the binomial formula, we calculated the likelihood of different scenarios in our exercise. This includes finding probabilities for exactly five believers, all six believers, and at least five believers among six adults. For instance, summing the probabilities for exactly five and six believers gives the likelihood of at least five believers:
\( P(X \ge 5) = P(X=5) + P(X=6) \approx 0.00154 + 0.000064 = 0.001604 \) < ul>
  • **Compare Data**: Compare calculated results with significance levels to understand their importance.
  • **Draw Conclusions**: Conclude accurate, data-backed interpretations.
    • Through these steps, statistical analysis helps us turn data into real-world insights.

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    Most popular questions from this chapter

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    Involve finding binomial probabilities, finding parameters, and determining whether values are significantly high or low by using the range rule of thumb and probabilities. The County Clerk in Essex, New Jersey, was accused of cheating by not using randomness in assigning line positions on voting ballots. Among 41 different ballots, Democrats were assigned the top line 40 times. Assume that Democrats and Republicans are assigned the top line using a method of random selection so that they are equally likely to get that top line. a. Use the range rule of thumb to identify the limits separating values that are significantly low and those that are significantly high. Based on the results, is the result of 40 top lines for Democrats significantly high? b. Find the probability of exactly 40 top lines for Democrats. c. Find the probability of 40 or more top lines for Democrats. d. Which probability is relevant for determining whether 40 top lines for Democrats is significantly high: the probability from part (b) or part (c)? Based on the relevant probability, is the result of 40 top lines for Democrats significantly high? e. What do the results suggest about how the clerk met the requirement of assigning the line positions using a random method?
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