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Chapter 5: Problem 87

The following table summarizes data on smoking status and age group, and is consistent with summary quantities obtained in a Gallup Poll published in the online article "In U.S., Young Adults' Cigarette Use Is Down Sharply" $$ \begin{array}{|lcc|} \hline & {\text { Smoking Status }} \\ \hline { 2 - 3 } \text { Age Group } & \text { Smoker } & \text { Nonsmoker } \\\ \hline 18 \text { to } 29 & 174 & 618 \\ 30 \text { to } 49 & 333 & 1,115 \\ 50 \text { to } 64 & 384 & 1,445 \\ 65 \text { and older } & 211 & 1,707 \\ \hline \end{array} $$ Assume that it is reasonable to consider these data as representative of the American adult population. Consider the chance experiment or randomly selecting an adult American. a. What is the probability that the selected adult is a smoker? b. What is the probability that the selected adult is under 50 years of age? c. What is the probability that the selected adult is a smoker that is 65 or older? d. What is the probability that the selected adult is a smoker or is age 65 or older?

Short Answer

Expert verified
Total = 4827 a. Smokers = 1102 P(Smoker) = \( \frac{1102}{4827} \) b. Under 50 = 2240 P(Under 50) = \( \frac{2240}{4827} \) c. Smokers 65+ = 211 P(Smoker 65+) = \( \frac{211}{4827} \) d. 65 and older = 1918 P(65 and older) = \( \frac{1918}{4827} \) P(Smoker or 65+) = \( \frac{1102}{4827} + \frac{1918}{4827} - \frac{211}{4827} \)

Step by step solution

01

Calculate the total population

To find probabilities, we first need to know the total number of adults in the given table. This is the sum of all the adults in every age group and smoking status: Total = (174 + 618) + (333 + 1115) + (384 + 1445) + (211 + 1707)
02

Calculate probabilities for each part of the question

a. To find the probability of an adult being a smoker, we need to calculate the number of smokers in the given data and divide that by the total population. The number of smokers is the sum of all the adults in the smoker column: Smokers = (174 + 333 + 384 + 211) P(Smoker) = Smokers / Total b. To find the probability of an adult being under 50, we need to calculate the number of adults in the age groups 18-29 and 30-49. This is the sum of both smoker and nonsmoker adults in these two age groups: Under 50 = (174 + 618) + (333 + 1115) P(Under 50) = Under 50 / Total c. To find the probability of an adult being a smoker aged 65 or older, we need to consider the number of smokers in the 65 and older age group: Smokers 65+ = 211 P(Smoker 65+) = Smokers 65+ / Total d. To find the probability of an adult being a smoker or aged 65 or older, we need to find the probability of an adult being a smoker, the probability of an adult being 65 or older, and the probability of an adult being a smoker aged 65 or older. Then, we can use the formula P(A or B) = P(A) + P(B) - P(A and B). First, we need to find the probability of an adult being 65 or older: 65 and older = (211 + 1707) P(65 and older) = 65 and older / Total P(Smoker or 65+) = P(Smoker) + P(65 and older) - P(Smoker 65+)

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Key Concepts

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

Representative Sample
Understanding the concept of a representative sample is essential for accurate data analysis. A representative sample is a subset of a population that accurately reflects the demographics or characteristics of the entire population. It's crucial when analyzing social data like smoking habits since it allows us to make inferences about the larger population from the sample data.

For example, the data drawn from the Gallup Poll in our exercise suggests it is reflective of the American adult population. This means that if we're looking at smoking status across different age groups, the sample must have a proportional number of adults from each age group corresponding to their distribution in the entire population.

When a sample is truly representative, the probability calculations derived from it can be indicative of real-world scenarios, giving insights into how often certain traits (like being a smoker) occur within the population at large.
Random Selection
Random selection is a core component in obtaining a representative sample and conducting probability calculations. It involves selecting individuals from the entire population in such a way that each member has an equal chance of being chosen. This method helps to minimize biases and ensures that the sample is not skewed towards any subset of the population.

In the context of the exercise, we consider the random selection of an American adult. This hypothetical process means that every adult in the population, regardless of their smoking status or age, has an equal chance of appearing in our sample data. Random selection is what makes the analysis of the data relevant to probability theory, as it aligns with the assumption that all outcomes are equally likely.
Probability Theory
Probability theory is a branch of mathematics that deals with the likelihood of events occurring. It provides a mathematical framework to quantify uncertainty and can predict outcomes when running the same scenario multiple times.

In our exercise, we employ probability theory to calculate the likelihoods of various scenarios relating to smoking status and age group. Probability is usually expressed as a fraction or percentage that ranges from 0 (impossible event) to 1 (certain event). To calculate probabilities, we use the relevant counts from our data (such as the number of smokers or those under 50 years old) and divide by the total number of individuals in the sample.
Smoking Status Data Analysis
Our exercise involves smoking status data analysis, which can provide insights into public health and demographic trends. We analyze the given data to determine probabilities of different smoking statuses across age groups.

Key steps include summing the data to find totals by category and then creating ratios to express the desired probabilities. For reliable results, we ensure that the data is first checked for being a representative sample. From this analysis, we can draw conclusions such as the proportion of smokers within specific age ranges and make population-wide estimates regarding smoking habits.
  • The probability of an adult being a smoker is found by dividing the total number of smokers by the total number of adults.
  • The probability of an adult being under 50 is calculated by adding together the adults in the 18-29 and 30-49 age groups and dividing by the total population.
  • For specific demographics like smokers 65 or older, we isolate the data from the relevant age and smoker status first before calculating the probability.
This form of analysis is useful for policymakers, health professionals, and researchers in understanding and addressing smoking-related issues within the population.

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Most popular questions from this chapter

The report "Twitter in Higher Education: Usage Habits and Trends of Today's College Faculty" describes a survey of nearly 2000 college faculty. The report indicates the following: \(30.7 \%\) reported that they use Twitter, and \(69.3 \%\) said that they do not use Twitter. Of those who use Twitter, \(39.9 \%\) said they sometimes use Twitter to communicate with students. Of those who use Twitter, \(27.5 \%\) said that they sometimes use Twitter as a learning tool in the classroom. Consider the chance experiment that selects one of the study participants at random. a. Two of the percentages given in the problem specify unconditional probabilities, and the other two percentages specify conditional probabilities. Which are conditional probabilities? How can you tell? b. Suppose the following events are defined: \(T=\) event that selected faculty member uses Twitter \(C=\) event that selected faculty member sometimes uses Twitter to communicate with students \(L=\) event that selected faculty member sometimes uses Twitter as a learning tool in the classroom Use the given information to determine the following probabilities: i. \(P(T)\) iii. \(P(C \mid T)\) ii. \(P\left(T^{C}\right)\) iv. \(\quad P(L \mid T)\) c. Construct a hypothetical 1000 table using the given probabilities and use the information in the table to calculate \(P(C),\) the probability that the selected study participant sometimes uses Twitter to communicate with students. d. Construct a hypothetical 1000 table using the given probabilities and use the information in the table to calculate the probability that the selected study participant sometimes uses Twitter as a learning tool in the classroom.

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