 Exercise 3.    A simple random sample (𝑆𝑅𝑆)  𝑠of 250 adults in Iowa City  was taken on June 29, 2023.  Each person’s  𝐷𝑖𝑠𝑒𝑎𝑠𝑒
 status (𝑦𝑒𝑠, 𝑛𝑜on July 1)  and 𝐸𝑥𝑝𝑜𝑠𝑒𝑑to smog status (𝑦𝑒𝑠,  𝑛𝑜) was recorded.   It turned out that of the 250 adults sampled,  
28  were diseased,  150 were exposed,  and 89  were neither diseased nor exposed.
 (a) Compute  𝑝𝑟𝑜𝑝 𝐷𝑖𝑠𝑒𝑎𝑠𝑒 𝑦𝑒𝑠 .
 (b) Compute  𝑝𝑟𝑜𝑝 𝐷𝑖𝑠𝑒𝑎𝑠𝑒 𝑦𝑒𝑠  𝐸𝑥𝑝𝑜𝑠𝑒𝑑 𝑦𝑒𝑠 .
 (c) Compute  𝑝𝑟𝑜𝑝 𝐷𝑖𝑠𝑒𝑎𝑠𝑒 𝑦𝑒𝑠  𝐸𝑥𝑝𝑜𝑠𝑒𝑑 𝑛𝑜 .
 (d) Sketch, or use software like Excel to create,  a side-by-side 100% stacked bar chart of the conditional distributions  
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�𝑖𝑠𝑡(𝐷𝑖𝑠𝑒𝑎𝑠𝑒:𝑠|𝐸𝑥𝑝𝑜𝑠𝑒𝑑 𝑦𝑒𝑠 and 𝑑𝑖𝑠𝑡𝐷𝑖𝑠𝑒𝑎𝑠𝑒:𝑠 |𝐸𝑥𝑝𝑜𝑠𝑒𝑑 𝑛𝑜 .  Clearly label your chart

### Given Data:

- Total sample size \( n = 250 \)
- Number of diseased individuals = 28
- Number of exposed individuals = 150
- Number of neither diseased nor exposed individuals = 89

We can now address the tasks:

### (a) Compute \( \text{prop Disease yes} \)

The proportion of individuals who are diseased is given by the formula:

\[
\text{prop Disease yes} = \frac{\text{Number of diseased individuals}}{\text{Total sample size}} = \frac{28}{250} = 0.112
\]

Thus, the proportion of diseased individuals is **0.112** or **11.2%**.

---

### (b) Compute \( \text{prop Disease yes | Exposed yes} \)

We need to find the proportion of diseased individuals among the exposed population. The number of exposed individuals is 150, and we know that 89 people were neither diseased nor exposed. Therefore, we can calculate the number of people who were either diseased or exposed but not both, using this information.

To find the number of people who are **both diseased and exposed**, we can subtract the number of people who are neither diseased nor exposed (89) from the total sample size:

\[
\text{Number of either diseased or exposed} = 250 - 89 = 161
\]

Since there are 150 people who were exposed and 28 diseased individuals, the number of people who are diseased and exposed can be derived as:

\[
\text{Diseased and Exposed} = 28 + 150 - 161 = 17
\]

Now, we can calculate the proportion of diseased individuals among the exposed population:

\[
\text{prop Disease yes | Exposed yes} = \frac{17}{150} \approx 0.1133
\]

Thus, the proportion is **0.1133** or **11.33%**.

---

### (c) Compute \( \text{prop Disease yes | Exposed no} \)

To find the number of people who are **diseased but not exposed**, we subtract the number of people who are diseased and exposed (17) from the total number of diseased individuals (28):

\[
\text{Diseased and Not Exposed} = 28 - 17 = 11
\]

Now, the number of people who are not exposed is \( 250 - 150 = 100 \). Therefore, the proportion of diseased individuals among those not exposed is:

\[
\text{prop Disease yes | Exposed no} = \frac{11}{100} = 0.11
\]

Thus, the proportion is **0.11** or **11%**.

---

### (d) Side-by-Side 100% Stacked Bar Chart

The data we have for the conditional distributions is:

- \( \text{prop Disease yes | Exposed yes} = 11.33\% \)
- \( \text{prop Disease no | Exposed yes} = 100\% - 11.33\% = 88.67\% \)
- \( \text{prop Disease yes | Exposed no} = 11\% \)
- \( \text{prop Disease no | Exposed no} = 100\% - 11\% = 89\% \)

#### Chart Sketch Description:
The 100% stacked bar chart should show two side-by-side bars:

1. **Exposed Yes**: 
   - 11.33% of the bar is colored to represent those with the disease, and the remaining 88.67% represents those without the disease.
   
2. **Exposed No**: 
   - 11% of the bar is colored to represent those with the disease, and the remaining 89% represents those without the disease.

Would you like me to generate this chart, or would you like help with it in Excel? 

---

### Follow-Up Questions:
1. How can the exposure-disease relationship be interpreted based on the results?
2. What assumptions are we making about the simple random sample (SRS)?
3. How might the sample size of 250 affect the precision of these proportions?
4. How could these results vary if the sample were taken at a different time of year?
5. What additional data could strengthen the analysis?

### Tip:
To interpret the significance of the proportions, comparing these conditional probabilities helps to understand the association between disease status and exposure to smog.

Exercise 3. A simple random sample (𝑆𝑅𝑆) 𝑠 of 250 adults in Iowa City was taken on June 29, 2023. Each person’s Disease status (𝑦𝑒𝑠, 𝑛𝑜 on July 1) and Exposed to smog status (𝑦𝑒𝑠, 𝑛𝑜) was recorded. It turned out that of the 250 adults sampled, 28 were diseased, 150 were exposed, and 89 were neither diseased nor exposed. (a) Compute 𝑝𝑟𝑜𝑝 𝐷𝑖𝑠𝑒𝑎𝑠𝑒 𝑦𝑒𝑠. (b) Compute 𝑝𝑟𝑜𝑝 𝐷𝑖𝑠𝑒𝑎𝑠𝑒 𝑦𝑒𝑠 𝐸𝑥𝑝𝑜𝑠𝑒𝑑 𝑦𝑒𝑠. (c) Compute 𝑝𝑟𝑜𝑝 𝐷𝑖𝑠𝑒𝑎𝑠𝑒 𝑦𝑒𝑠 𝐸𝑥𝑝𝑜𝑠𝑒𝑑 𝑛𝑜. (d) Sketch, or use software like Excel to create, a side-by-side 100% stacked bar chart of the conditional distributions Dist(𝐷𝑖𝑠𝑒𝑎𝑠𝑒:𝑠|𝐸𝑥𝑝𝑜𝑠𝑒𝑑 𝑦𝑒𝑠 and Dist(𝐷𝑖𝑠𝑒𝑎𝑠𝑒:𝑠 |𝐸𝑥𝑝𝑜𝑠𝑒𝑑 𝑛𝑜). Clearly label your chart.

Learn how to compute proportions and conditional probabilities from a random sample of 250 adults in Iowa City. This problem explores the relationships between disease status and exposure to smog using statistical concepts and probability formulas. Includes side-by-side 100% stacked bar charts. Suitable for college-level statistics.

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