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%.

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(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%.

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(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%.

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(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?

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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.