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Chapter 5: Problem 43

A nationwide survey of 17,000 seniors by the University of Michigan revealed that almost \(70 \%\) disapprove of daily pot smoking. If 18 of these seniors are selected at random and asked their opinions, what is the probability that more than 9 but less than 14 disapprove of smoking pot?

Short Answer

Expert verified
The probability that more than 9 but less than 14 seniors disapprove of smoking pot equals the summation of Binomial probabilities for 10, 11, 12 and 13 successes

Step by step solution

01

Understand the Binomial Probability Formula

The binomial probability formula is given by \( P_r = C(n, r) * (p^r) * ( 1 - p )^{( n - r )} \) where \( P_r \) represents probability with r successes, n is the number of trials and p is the probability of success in a single trial.
02

Calculation of Parameters

From the question, we have the following parameters: Total trials n = 18, Success on a single trial p = 0.7. Now replace the parameters in the binomial formula.
03

Calculation Of Binomial Probabilities

We want to know how likely it is to have more than 9 but less than 14 disapprovals. This means we calculate the binomial probabilities for 10, 11, 12 and 13 successes (disapprovals) separately and then sum the results.
04

Summation Of Results

Calculate the binomial probabilities separately using the binomial formula. Then add all the results together to get the total probability.

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Key Concepts

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

Probability Theory
Probability theory is a fundamental part of mathematics that helps us gauge the chance of occurrence of various events. Probability is a number between 0 and 1, where 0 means the event will not occur, and 1 means the event will certainly happen. This theory allows us to make predictions about future events based on existing data or repeating experiments.

Key elements include:
  • Random Experiment: Any situation where there is uncertainty about the outcome can be termed a random experiment.
  • Sample Space: The set of all possible outcomes is called the sample space.
  • Event: Any subset of a sample space is called an event, and events can be simple or compound.
In the context of survey results like the one in the exercise, probability theory is applied to predict the likelihood of certain survey responses within a sample group, based on a known proportion of such responses in a larger population.
Binomial Probability Formula
The binomial probability formula is a key component when dealing with events where there are two possible outcomes, often termed as "success" or "failure." In our survey example, a "success" is defined as a senior disapproving of smoking pot on a daily basis. The formula is structured as: \[ P_r = \binom{n}{r} \cdot p^r \cdot (1-p)^{n-r} \]Where:
  • \( P_r \) = Probability of getting exactly \( r \) successes in \( n \) trials.
  • \( \binom{n}{r} \) = A combination function indicating the number of ways to choose \( r \) successes out of \( n \) trials.
  • \( p \) = Probability of success on an individual trial (0.7 in our case).
  • \( (1-p) \) = Probability of failure on an individual trial.
The binomial distribution is particularly useful for survey analysis as it allows us to compute the probability of a certain number of participants in a sample having a common characteristic, such as a specific opinion or behavior, based on the overall likelihood in the population.
Statistical Survey Analysis
Statistical survey analysis involves applying statistical tools to interpret data collected from surveys. This analysis helps in making informed decisions and predictions. Surveys often seek to estimate probabilities about the behaviors, opinions, or characteristics of a larger population.

Key steps include:
  • Sample Selection: Carefully choosing a representative group from the broader population to reflect broader trends accurately.
  • Data Collection: Gathering responses through structured questions aimed at obtaining clear and usable data.
  • Data Analysis: Employing probability distributions like the binomial distribution to make inferences about the entire population based on the sample.
In our exercise, survey analysis aids in predicting how many seniors disapprove of daily pot smoking when randomly selecting 18 individuals. We calculate probabilities for different levels of disapproval, helping researchers understand likely outcomes in the group and extend those findings to a larger population.

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Most popular questions from this chapter

An electronic switching device occasionally malfunctions and may need to be replaced. It is known that the device is satisfactory if it makes, on the average, no more than 0.20 error per hour. A particular 5-hour period is chosen as a "test" on the device. If no more than 1 error occurs, the device is considered satisfactory. (a) What is the probability that a satisfactory device will be considered unsatisfactory on the basis of the rest? Assume that a Poisson process exists. (b) What is the probability that a device will be accepted as satisfactory when, in fact, the mean number of errors is \(0.25 ?\) Again, assume that a Poisson process exists.

A foreign student club lists as its members 2 Canadians, 3 Japanese, 5 Italians, and 2 Germans. If a committee of 4 is selected at random, find the probability that (a) all nationalities are represented; (b) all nationalities except the Italians are represented.

It is estimated that 4000 of the 10,000 voting residents of a town are against a new sales tax. If 15 eligible voters are selected at random and asked their opinion, what is the probability that at most 7 favor the new tax?

A company purchases large lots of a certain kind of electronic device, A method is used that rejects a lot if 2 or more defective units are found in a random sample of 100 units. (a) What is the mean number of defective units found in a sample of 100 units if the lot is \(1 \%\) defective? (b) What is the variance?

If the probability that a fluorescent light has a useful life of at least 800 hours is \(0.9,\) find the probabilities that among 20 such lights (a) exactly 18 will have a useful life of at least 800 hours; (b) at. least 15 will have a useful life of at least 800 hours; (c) at least. 2 will not have a useful life of at least 800 hours.
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