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12.4.3 Quiz: Two-Way Frequency Tables

This table shows the results of a study on side effects:

\begin{tabular}{|l|c|c|c|}
\hline
& Side effects & No side effects & Total \\
\hline
Adults & 7 & 43 & 50 \\
\hline
Children & 22 & 28 & 50 \\
\hline
Total & 29 & 71 & 100 \\
\hline
\end{tabular}

Compare the probability that an adult has side effects with the probability that a child has side effects. Draw a conclusion based on your results.

A. [tex] P(\text{side effects} \mid \text{child}) = 0.22 \]
[tex] P(\text{side effects} \mid \text{adult}) = 0.70 \]
Conclusion: Children have a much lower chance of having side effects than adults.

B. [tex] P(\text{side effects} \mid \text{child}) = 0.44 \]
[tex] P(\text{side effects} \mid \text{adult}) = 0.14 \]
Conclusion: Children have a much greater chance of having side effects than adults.

C. [tex] P(\text{side effects} \mid \text{child}) = 0.22 \]
[tex] P(\text{side effects} \mid \text{adult}) = 0.70 \]
Conclusion: Children have a much greater chance of having side effects than adults.

D. [tex] P(\text{side effects} \mid \text{child}) = 0.44 \]
[tex] P(\text{side effects} \mid \text{adult}) = 0.14 \]
Conclusion: Children have a much lower chance of having side effects than adults.


Sagot :

To compare the probabilities of side effects between adults and children, let's break down the steps thoroughly:

1. Identify the relevant data:
- Number of adults with side effects: 7
- Number of adults without side effects: 43
- Number of children with side effects: 22
- Number of children without side effects: 28
- Total number of adults: 50 (7 + 43)
- Total number of children: 50 (22 + 28)

2. Calculate the probability of side effects for adults:
[tex]\[ P(\text{side effects} \mid \text{adult}) = \frac{\text{Number of adults with side effects}}{\text{Total number of adults}} \][/tex]
Substituting the values, we get:
[tex]\[ P(\text{side effects} \mid \text{adult}) = \frac{7}{50} = 0.14 \][/tex]

3. Calculate the probability of side effects for children:
[tex]\[ P(\text{side effects} \mid \text{child}) = \frac{\text{Number of children with side effects}}{\text{Total number of children}} \][/tex]
Substituting the values, we get:
[tex]\[ P(\text{side effects} \mid \text{child}) = \frac{22}{50} = 0.44 \][/tex]

4. Comparison and conclusion:
- Probability of side effects for adults: [tex]\(0.14\)[/tex]
- Probability of side effects for children: [tex]\(0.44\)[/tex]

Since [tex]\(0.44\)[/tex] (probability for children) is greater than [tex]\(0.14\)[/tex] (probability for adults), we conclude that children have a greater chance of having side effects compared to adults.

Therefore, the correct option is:

B.
[tex]\[ P(\text{side effects} \mid \text{child}) = 0.44 \][/tex]
[tex]\[ P(\text{side effects} \mid \text{adult}) = 0.14 \][/tex]
Conclusion: Children have a much greater chance of having side effects than adults.
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