Problem 44 - Applied Mathematics: For the Managerial, Life, and Social Sciences

Question

Researchers weighed 1976 3-yr-olds from low-income families in 20 U.S. cities. Each child is classified by race (white, black, or Hispanic) and by weight (normal weight, overweight, or obese). The results are tabulated as follows: $$\begin{array}{lcccc}\hline & &{\text { Weight, \% }} & \\ \text { Race } & \text { Children } & \text { Normal Weight } & \text { Overweight } & \text { Obese } \\\\\hline \text { White } & 406 & 68 & 18 & 14 \\ hline \text { Black } & 1081 & 68 & 15 & 17 \\ \hline \text { Hispanic } & 489 & 56 & 20 & 24 \\ \hline\end{array}$$ If a participant in the research is selected at random and is found to be obese, what is the probability that the 3 -yr-old is white? Hispanic?

Step-by-Step Solution

Answer

The probability that a randomly selected obese child is White is approximately \(16.0\% \), and the probability that a randomly selected obese child is Hispanic is approximately \(32.6\% \).

Step 1: Find the marginal probabilities of each race

To find the total number of children who are White, Black, or Hispanic, we can simply add the number of children in each category in the given table. We will use these probabilities to calculate the joint probability of selecting a child from a particular race who is obese.

Step 2: Calculate the joint probability of selecting an obese child

To find the joint probability of selecting an obese child, we first need to determine the total number of obese children. This can be found by multiplying the percentage of obese children for each race by the total number of children in that racial category, and then dividing by the total number of children (1976): Total obese children = (0.14 * 406) + (0.17 * 1081) + (0.24 * 489) Total obese children = 57.24 + 183.77 + 117.36 Total obese children ≈ 358 Now, we have the total number of obese children, so we can calculate the probability of selecting an obese child: P(Obese) = Total obese children / Total children P(Obese) = 358 / 1976 P(Obese) ≈ 0.181

Step 3: Calculate the joint probability of selecting an obese child who is White or Hispanic

To calculate the joint probability of selecting an obese child who is White or Hispanic, we will use the percentage of obese children for each race and multiply it by the total number of children in that racial category, then divide by the total number of children (1976): P(White and Obese) = (0.14 * 406) / 1976 ≈ 57.24 / 1976 ≈ 0.029 P(Hispanic and Obese) = (0.24 * 489) / 1976 ≈ 117.36 / 1976 ≈ 0.059

Step 4: Calculate the conditional probabilities

Now we can calculate the conditional probabilities: P(White | Obese) = P(White and Obese) / P(Obese) P(White | Obese) = 0.029 / 0.181 ≈ 0.160 P(Hispanic | Obese) = P(Hispanic and Obese) / P(Obese) P(Hispanic | Obese) = 0.059 / 0.181 ≈ 0.326 So, the probability that a randomly selected obese child is White is approximately 16.0%, and the probability that a randomly selected obese child is Hispanic is approximately 32.6%.

Key Concepts

Race and Obesity Statistics

Statistics on race and obesity can offer valuable insights into public health issues, and this data can help inform targeted interventions for specific communities. The exercise at hand involves a study on 3-year-olds from low-income families in 20 different U.S. cities. The children are categorized based on race: White, Black, or Hispanic, and their weight status: normal weight, overweight, or obese.

Understanding the prevalence of obesity among different races within these young children is essential. In this case, the data shows:

• 68% of White children have a normal weight, 18% are overweight, and 14% are obese.
• 68% of Black children have a normal weight, 15% are overweight, and 17% are obese.
• 56% of Hispanic children have a normal weight, 20% are overweight, and 24% are obese.

This breakdown serves as a foundation for calculating probabilities related to obesity and race, which is crucial for understanding how these factors intersect.

Probability Calculation Steps

Calculating probability involves several steps to ensure accurate results, especially when dealing with complex statistics such as race and obesity. Here, we will walk through the steps to determine the probability of selecting a child of a certain race, given that the child is obese.

Firstly, we begin with marginal probabilities, which means calculating the total number of children in each racial category. This helps us understand the broader context of the study's demographics.

Next, the joint probability is calculated for selecting an obese child. This means determining how many children are obese overall. We do this by multiplying the percentage of obese children for each race by the total children in that race:

• White obese children: \[(0.14 \times 406) = 57.24\]
• Black obese children: \[(0.17 \times 1081) = 183.77\]
• Hispanic obese children: \[(0.24 \times 489) = 117.36\]
• Total obese children: \[57.24 + 183.77 + 117.36 \approx 358\]

Finally, we calculate the probability of a child being both obese and from a certain race, such as White or Hispanic. This gives us the joint probabilities required for the next step.

Marginal and Joint Probabilities

Understanding the difference between marginal and joint probabilities is key to solving this exercise. **Marginal probability** refers to the probability of a single event occurring without regard to any other factors. In our exercise, it is the likelihood of an obese child being chosen from the entire group of children, regardless of race:

\[ P(\text{Obese}) = \frac{358}{1976} \approx 0.181 \] or about 18.1%.

**Joint probability** comes into play when calculating the likelihood of two events occurring together. Here we focus on the probability of a child being both obese and a member of a specific race (e.g., White or Hispanic):

• Probability of a child being both White and Obese:\[ P(\text{White and Obese}) = \frac{57.24}{1976} \approx 0.029 \], or 2.9%.
• Probability of a child being both Hispanic and Obese:\[ P(\text{Hispanic and Obese}) = \frac{117.36}{1976} \approx 0.059 \], or 5.9%.

These calculations feed into the conditional probability, which tells us the likelihood of a child being a specific race, given they are obese.