Problem 75 - Precalculus

Question

On January 6 and \(10,2000,\) the Harris Poll conducted a survey of adult smokers in the United States. When asked, "Have you ever tried to quit smoking?", \(70 \%\) said yes and \(30 \%\) said no. Write a \(2 \times 1\) matrix- call it \(A\) - that represents those smokers. When asked what consequences smoking would have on their lives, \(89 \%\) belicved it would increase their chance of getting lung cancer and \(84 \%\) belicved smoking would shorten their lives. Write a \(2 \times 1\) matrix - call it \(B\) - that represents those smokers. If there are 46 million adult smokers in the United States a. What does \(46 A\) tell us? b. What does \(46 B\) tell us?

Step-by-Step Solution

Answer

46A tells us about the number of smokers trying to quit, 46B tells us about beliefs in smoking consequences.

Step 1: Understanding the matrices

We need to create two \(2 \times 1\) matrices, \(A\) and \(B\). Matrix \(A\) will represent the percentage of smokers who have tried to quit and those who haven't. Matrix \(B\) will represent the percentage of smokers who believe smoking has certain health effects.

Step 2: Construct Matrix A

Matrix \(A\) represents the smokers' responses to whether they have tried to quit. It will look like this:\[ A = \begin{bmatrix} 70 \ 30 \end{bmatrix} \]This matrix indicates that 70% have tried to quit and 30% haven't.

Step 3: Construct Matrix B

Matrix \(B\) represents smokers' beliefs about the health consequences of smoking. It will be written as:\[ B = \begin{bmatrix} 89 \ 84 \end{bmatrix} \]The matrix shows that 89% believe smoking increases lung cancer risk, and 84% believe it shortens life span.

Step 4: Calculate 46A

To find \(46A\), multiply each entry of matrix \(A\) by 46 million. This represents the number of smokers in each category:\[46A = 46 \times \begin{bmatrix} 70 \ 30 \end{bmatrix} = \begin{bmatrix} 46 \times 70 \ 46 \times 30 \end{bmatrix} = \begin{bmatrix} 3220 \ 1380 \end{bmatrix}\]This calculation shows there are 32.2 million smokers who have tried to quit and 13.8 million who haven't.

Step 5: Calculate 46B

Next, compute \(46B\) by multiplying each entry of matrix \(B\) by 46 million to know the number of smokers who believe in each consequence:\[46B = 46 \times \begin{bmatrix} 89 \ 84 \end{bmatrix} = \begin{bmatrix} 46 \times 89 \ 46 \times 84 \end{bmatrix} = \begin{bmatrix} 4094 \ 3864 \end{bmatrix}\]This indicates that 40.94 million smokers believe smoking increases lung cancer risk, and 38.64 million believe it shortens life.

Key Concepts

Matrices

Matrices are not just a fancy tool in mathematics, they are a powerful way to organize and handle data. Essentially, a matrix is a rectangular array of numbers arranged in rows and columns. In this exercise, we are dealing with a special type of matrix called a column matrix or column vector because it has only one column. You might see matrices like this all around, especially when dealing with lots of data that need compact representation, like survey results or statistical data.

• In this context, we use the matrix \(A\) to represent the percentage of smokers who have tried to quit and those who haven't. Each entry in the matrix provides a piece of this survey's story.
• Likewise, matrix \(B\) expresses smokers' beliefs about the health consequences of smoking. Every entry tells us a proportion of people sharing a common belief.

Visualizing data in this way makes it easier to see relationships and draw conclusions from numerous numbers.

Percentage Calculations

Understanding percentages is crucial when dealing with survey results. Percentages offer a way to express data in a form that's easy to comprehend. For example, saying that 70% of smokers have attempted to quit is more intuitive than simply giving an unconnected figure.

• Each matrix we work with in this scenario—\(A\) and \(B\)—contains entries representing percentages. These figures tell us how survey respondents are divided among different opinions.
• When we calculate, say, \(46A\), we are effectively converting those percentages into absolute numbers by using the total number of smokers, which is 46 million in this case. Multiplying the percentage by the total provides an estimate of the actual number of people.

This conversion from percentages to real numbers helps vividly demonstrate the size of each section of the survey population, offering tangible insights into the data.

Statistics

Statistics play a crucial role in understanding surveys. They help us make sense of data at a glance and draw conclusions from it. By gathering data from smokers on their beliefs and actions, the Harris Poll used a statistical approach to analyze public opinions.

• The calculation of percentages and conversion into numbers of smokers (e.g., calculating \(46A\) and \(46B\)) is a basic statistical method used to interpret survey data.
• The results give insights into prevailing attitudes and behaviors among smokers, which can be significant for policy-making or public health campaigns.

Through statistics, large datasets become comprehensible, and they give us the tools to ask deeper questions and probe for more meaningful answers.

Smoking Survey Analysis

Analyzing a smoking survey like this one helps us discern patterns and trends among smokers. It offers a snapshot of their behavior and beliefs at a specific time point.

• When asked about quitting, 70% saying 'yes' and 30% saying 'no' tells us there is a strong inclination toward cessation attempts, potentially pointing to awareness about health risks.
• The follow-up question about health consequences shows a significant majority believing in serious health impacts—89% about lung cancer risk and 84% about life-shortening effects.
• By interpreting these results with matrices, we not only uncover the proportion but also visualize this in the form of actual numbers, making the data actionable.

Such surveys and analyses provide valuable insights for health organizations looking to understand and assist community efforts in the fight against smoking.