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Chapter 4: Problem 7

A recent Harris Poll survey of 1010 U.S. adults selected at random showed that 627 consider the occupation of firefighter to have very great prestige. Estimate the probability (to the nearest hundredth) that a U.S. adult selected at random thinks the occupation of firefighter has very great prestige.

Short Answer

Expert verified
The probability is approximately 0.62.

Step by step solution

01

Understand the Data

We need to determine the probability that a randomly selected U.S. adult thinks highly of the occupation of firefighter. The survey provides us with 1010 total adults and 627 adults who view the occupation that way.
02

Set Up the Probability Formula

Probability is calculated as the number of favorable outcomes over the total number of outcomes. Here, the favorable outcomes are the adults who find firefighter occupation prestigious, while the total outcomes are the total surveyed individuals.
03

Apply the Probability Formula

To find the probability, divide the number of adults who see firefighters as prestigious by the total number of surveyed adults: \( \frac{627}{1010} \).
04

Calculate the Decimal Value

Perform the division to find the probability in decimal form: \( \frac{627}{1010} \approx 0.62079 \).
05

Round the Answer

Round the result from Step 4 to the nearest hundredth. This gives us the probability \( 0.62 \).

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Key Concepts

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

Survey Analysis
Survey analysis is a crucial part of understanding public opinion through data collection and interpretation. In this context, a survey was conducted to understand how U.S. adults perceive the prestige of certain professions, like firefighters. Let鈥檚 look at why surveys are important:
  • **Data Collection**: Surveys gather data from a group that represents a larger population. Here, 1010 U.S. adults were asked about their thoughts on firefighter prestige.
  • **Random Selection**: Participants were randomly chosen to minimize bias. Randomness ensures that every individual in the wider population had an equal chance of being surveyed.
  • **Quantitative Insight**: Surveys provide numerical data, which can be analyzed to draw conclusions about how the wider population might think or behave.
By analyzing the survey responses, we can estimate how many U.S. adults might hold firefighters in high esteem. This estimation process involves calculating probabilities, helping us translate survey data into meaningful predictions about the general public.
Rounding Numbers
When dealing with probabilities in survey data, the results often come out to multiple decimal places. Rounding numbers is used to make data easier to interpret and communicate. Here鈥檚 why and how we round in probability calculations:
  • **Simplification**: It makes complex numbers easy to understand and communicate. In our calculation, rounding simplifies the result from 0.62079 to 0.62.
  • **Precision vs. Understandability**: While precise numbers are accurate, they can overwhelm or confuse the audience. We round to the nearest hundredth place to balance precision with ease of understanding.
  • **Standard Practice**: Rounding to two decimal places (hundredths) is common in statistics as it provides a reasonable degree of accuracy without unnecessary details.
In the firefighter prestige survey, we rounded the probability from 0.62079 to 0.62, thus making the results more digestible and easy to relay to others without losing significant informational value.
Statistical Estimation
Statistical estimation involves using sample data to make generalizations about a population. In this survey, we want to estimate how people across the entire U.S. view the prestige of firefighters based on our sample.
  • **Sample to Population**: We use sample results to predict broader trends. Here, 627 out of 1010 respondents viewed firefighters positively, which we estimate to reflect the whole population.
  • **Probability**: A key measure in statistical estimation is probability. We calculated a probability of 0.62, estimating that 62% of the U.S. adult population hold firefighters in high regard.
  • **Assumptions and Limitations**: Estimations assume that the sample is representative of the broader population. However, there are limitations like sampling errors that might affect accuracy.
Statistical estimation converts the survey's sample results into a meaningful picture of public opinion. By doing so, it helps stakeholders like policymakers or businesses to make informed decisions.

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Most popular questions from this chapter

Medical: Tuberculosis The state medical school has discovered a new test for tuberculosis. (If the test indicates a person has tuberculosis, the test is positive.) Experimentation has shown that the probability of a positive test is 0.82, given that a person has tuberculosis. The probability is 0.09 that the test registers positive, given that the person does not have tuberculosis. Assume that in the general population, the probability that a person has tuberculosis is 0.04. What is the probability that a person chosen at random will (a) have tuberculosis and have a positive test? (b) not have tuberculosis? (c) not have tuberculosis and have a positive test?

In the Cash Now lottery game there are 10 finalists who submitted entry tickets on time. From these 10 tickets, three grand prize winners will be drawn. The first prize is \(\$ 1 dollars million, the second prize is \)\$ 100,000 dollars and the third prize is \(\$ 10,000 .\) Determine the total number of different ways in which the winners can be drawn. (Assume that the tickets are not replaced after they are drawn.)

General: Candy Colors \(\mathrm{M} \& \mathrm{M}\) plain candies come in various colors. The distribution of colors for plain M\&M candies in a custom bag is $$\begin{array}{l|ccccccc} \hline \text { Color } & \text { Purple } & \text { Yellow } & \text { Red } & \text { Orange } & \text { Green } & \text { Blue } & \text { Brown } \\ \hline \text { Percentage } & 20 \% & 20 \% & 20 \% & 10 \% & 10 \% & 10 \% & 10 \% \\ \hline \end{array}$$ Suppose you have a large custom bag of plain M\&M candies and you choose one candy at random. Find (a) \(P\) (green candy or blue candy). Are these outcomes mutually exclusive? Why? (b) \(P\) (yellow candy or red candy). Are these outcomes mutually exclusive? Why? (c) \(P(\text {not purple candy}).\)

If two events are mutually exclusive, can they occur concurrently? Explain.

You roll two fair dice, a green one and a red one. (a) Are the outcomes on the dice independent? (b) Find \(P(1 \text { on green die and } 2 \text { on red die })\) (c) Find \(P(2 \text { on green die and } 1\) on red die). (d) Find \(P[(1 \text { on green die and } 2 \text { on red die) or }(2 \text { on green die and } 1\text { on }\) red die)].
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