Problem 12 - Applied Mathematics: For the Managerial, Life, and Social Sciences

Question

In an online survey of 500 adults living with children under the age of \(18 \mathrm{yr}\), the participants were asked how many days per week they cook at home. The results of the survey are summarized below: $$ \begin{array}{lcccccccc} \hline \text { Number of Days } & 0 & 1 & 2 & 3 & 4 & 5 & 6 & 7 \\ \hline \text { Respondents } & 25 & 30 & 45 & 75 & 55 & 100 & 85 & 85 \\ \hline \end{array} $$ Determine the empirical probability distribution associated with these data.

Step-by-Step Solution

Answer

The empirical probability distribution associated with these data is: - P(0 days) = \(0.05\) - P(1 day) = \(0.06\) - P(2 days) = \(0.09\) - P(3 days) = \(0.15\) - P(4 days) = \(0.11\) - P(5 days) = \(0.20\) - P(6 days) = \(0.17\) - P(7 days) = \(0.17\)

Step 1: Identify the total number of respondents

We are given that there are 500 adults in the survey: Total respondents = 500

Step 2: Calculate the probability for each number of days

Divide the number of respondents for each number of days by the total number of respondents, and then represent it as a fraction and a percentage.

Step 3: Create a probability distribution

Use the values found in step 2 to create a probability distribution. \[ \begin{array}{lccccccccc} \hline \text{Number of Days} & 0 & 1 & 2 & 3 & 4 & 5 & 6 & 7 \\ \text{Respondents} & 25 & 30 & 45 & 75 & 55 & 100 & 85 & 85 \\ \text{Probability} & \frac{25}{500} & \frac{30}{500} & \frac{45}{500} & \frac{75}{500} & \frac{55}{500} & \frac{100}{500} & \frac{85}{500} & \frac{85}{500} \\ & 0.05 & 0.06 & 0.09 & 0.15 & 0.11 & 0.20 & 0.17 & 0.17 \\ \hline \end{array} \] The empirical probability distribution associated with these data is: - P(0 days) = 0.05 - P(1 day) = 0.06 - P(2 days) = 0.09 - P(3 days) = 0.15 - P(4 days) = 0.11 - P(5 days) = 0.20 - P(6 days) = 0.17 - P(7 days) = 0.17

Key Concepts

Probability Distribution

Understanding the concept of a probability distribution is essential for analyzing data and making predictions. Simply put, a probability distribution is a function that represents the likelihood of various possible outcomes in an experiment or a random event. In our exercise, we look at the number of days adults cook at home per week based on survey data.

The probability of each outcome is calculated by dividing the number of times a particular event occurs (for example, cooking 0 days a week) by the total number of observations (in this case, 500 survey respondents). The result shows us what fraction of the time we can expect each event to occur. For instance, if we randomly picked one of the surveyed adults, there is a 5% chance, or \( P(0 \text{ days}) = 0.05 \), that this adult does not cook at home at all in a week.

These probabilities are useful because they can help us predict future events or understand patterns within a set of data. For example, if a grocery store wanted to target adults who cook at home frequently for promotions, this probability distribution would suggest focusing on individuals who cook 5 to 7 days a week, as these categories have higher probabilities.

Data Analysis

Data analysis is the process of examining, cleaning, transforming, and modeling data with the aim of discovering useful information that supports decision-making. When analyzing survey data, as in our exercise, we first summarize the raw data to make it more digestible. This involves calculating frequencies or percentages, which are then used to create visual representations like graphs or charts, or to calculate statistical measures such as mean, median or probability distributions.

After establishing the empirical probability distribution for the number of days adults in the survey cooked at home, we can use it to gain insights into the population's cooking habits. For instance, the distribution shows a higher probability for cooking at home 5 to 7 days a week. This piece of information could be crucial for businesses like grocery stores or companies providing meal prep solutions, as it highlights potential market segments.

Therefore, effective data analysis not only helps with understanding past behaviors but also with anticipating future trends. It allows businesses, policymakers, and individuals to make informed, data-driven decisions.

Survey Data

Survey data is a type of data that is collected directly from respondents. It is often collected for research purposes to understand opinions, behaviors, or demographics of a particular group. In our exercise, we have survey data from 500 adults who report their home cooking frequency. One key aspect of working with survey data is ensuring the sample size is large enough to be representative of the larger population, thus providing more accurate results.

To enhance understanding and draw more precise conclusions from survey data, it's crucial to consider factors like sampling methods, survey design, and response rates. A well-structured survey can minimize biases and inaccuracies. Moreover, privacy considerations must be addressed when collecting and sharing survey data, especially when it includes personal information.

In conclusion, survey data gives us valuable insights into the behaviors and preferences of specific groups. When analyzed correctly, it can significantly contribute to various fields, from social sciences to market research, helping to shape products, services, and public policies.