Problem 39 - A Graphical Approach to College Algebra

Question

The table gives the results of a survey of \(14,000\) college students who were cigarette smoker in a recent year. \begin{array}{l|c}\hline \text { Number of Cigarettes per Day } & \text { Percent (as a Decimal) } \\\\\hline \text { Less than 1 } & 0.45 \\\1 \text { to } 9 & 0.24 \\\10 \text { to } 19 & 0.20 \\\\\text { A pack of 20 or more } & 0.11\end{array} Using the percents as probabilities, approximate the probability that, out of 10 of these student smokers selected at random, the following were true. Fewer than 2 smoked between 1 and 19 cigarettes per day.

Step-by-Step Solution

Answer

The probability is approximately 0.1445.

Step 1: Identify Probabilities

First, identify the probabilities of interest from the survey data. The probability that a student smokes between 1 and 9 cigarettes per day is \( P(A) = 0.24 \). The probability that a student smokes between 10 and 19 cigarettes per day is \( P(B) = 0.20 \). Therefore, the probability that a student smokes between 1 and 19 cigarettes per day is \( P(A \text{ or } B) = 0.24 + 0.20 = 0.44 \).

Step 2: Define 'Fewer than 2' Criteria

Since we want to know the probability that fewer than 2 students out of 10 smoke between 1 and 19 cigarettes per day, we use the binomial distribution where \( n = 10 \) and \( p = 0.44 \). We are interested in the cases where 0 or 1 student smokes between 1 and 19 cigarettes.

Step 3: Calculate Binomial Probabilities

For a binomial distribution, the probability of \( k \) successes in \( n \) trials is given by the formula: \( P(X = k) = \binom{n}{k} p^k (1-p)^{n-k} \).Calculate \( P(X = 0) \):\[ P(X = 0) = \binom{10}{0} (0.44)^0 (0.56)^{10} = 1 \times 1 \times 0.0075 = 0.0075 \]Calculate \( P(X = 1) \):\[ P(X = 1) = \binom{10}{1} (0.44)^1 (0.56)^9 = 10 \times 0.44 \times 0.0311 = 0.137 \]

Step 4: Compute Total Probability

Add the probabilities calculated in the previous step to find the total probability that fewer than 2 students smoke between 1 and 19 cigarettes per day.\[ P(X < 2) = P(X = 0) + P(X = 1) = 0.0075 + 0.137 = 0.1445 \]

Key Concepts

Binomial Distribution

The binomial distribution is a vital concept in statistics that describes the outcomes of a yes-or-no trial that is repeated multiple times. In our example, we are interested in situations where each college student smoker is considered a trial. Specifically, each student is asked whether they smoke between 1 to 19 cigarettes per day.

Binomial distribution requires two main parameters:

• \( n \): the number of trials or students being surveyed, which in this case is 10.
• \( p \): the probability of "success" on each trial. Here, a "success" is a student smoking between 1 and 19 cigarettes a day, which is 0.44 based on survey data.

A critical characteristic of a binomial distribution is that each trial is independent. This means the smoking habit of one student does not influence another. Also, the probability of success (\( p \)) is constant across trials.

Understanding binomial distribution helps us calculate the probability for different numbers of students, such as zero or one student smoking between 1 and 19 cigarettes, using the provided formulae.

College Student Surveys

Surveys of college students can provide valuable insights into health-related behaviors, like cigarette smoking. In such surveys, large groups of students, in this case, \( 14,000 \), are asked questions to gather data on their habits.

From this data, we see a detailed breakdown of cigarette consumption:

• 45% of students smoke less than 1 cigarette a day.
• 24% smoke between 1 and 9 cigarettes daily.
• 20% smoke between 10 and 19 cigarettes daily.
• 11% smoke a pack of 20 or more cigarettes a day.

Surveys are a snapshot of the habits during a particular period and can be used to track changes over time. They also help in identifying risk groups. For college students, the focus is often on behaviors affecting health and academic performance. Such surveys rely on both random sampling and self-reported data, making them powerful tools despite the potential for bias.

Cigarette Smoking Statistics

Understanding cigarette smoking statistics helps frame public health efforts to reduce smoking. Statistical data from surveys can inform anti-smoking campaigns and policy changes aimed at decreasing smoking rates.

According to the survey data we have, 55% of surveyed college students smoke at least some cigarettes daily. By assessing the range of cigarettes smoked each day, health officials can better target interventions and support groups for students who smoke the most. The identification of patterns in smoking prevalence helps guide tailored prevention and cessation programs aimed at young adults.

Statistics, like the percentages found in our survey, play an essential role not only in understanding behaviors but also in predicting future trends, which is crucial for planning healthcare resources and educational campaigns. The focus on the changeable habits of students is particularly important since habits developed during college years can persist into adulthood.