Sure! Let’s solve **part (a)** first, step-by-step, as requested: ### **Step 1: Organize the Data** From the table: | | Throat Cancer (Cases) | No Throat Cancer (Controls) | |------------------------|----------------------|-----------------------------| | Ever smoked | 40 | 20 | | Never smoked | 60 | 80 | | Total | 100 | 100 | --- ### **Step 2: Calculate the Odds of Exposure (Ever Smoked) in Each Group** - **Odds among cases (throat cancer):** \[ \text{Odds}_{\text{cases}} = \frac{\text{Ever smoked (cases)}}{\text{Never smoked (cases)}} = \frac{40}{60} = \frac{2}{3} \] - **Odds among controls (no throat cancer):** \[ \text{Odds}_{\text{controls}} = \frac{\text{Ever smoked (controls)}}{\text{Never smoked (controls)}} = \frac{20}{80} = \frac{1}{4} \] --- ### **Step 3: Calculate the Odds Ratio (OR)** \[ \text{Odds Ratio (OR)} = \frac{\text{Odds}_{\text{cases}}}{\text{Odds}_{\text{controls}}} = \frac{\frac{2}{3}}{\frac{1}{4}} = \frac{2}{3} \times \frac{4}{1} = \frac{8}{3} \approx 2.67 \] **Explanation:** The odds ratio compares the odds of exposure (ever smoking) among cases versus controls. Here, the OR of ≈ 2.67 means that the odds of ever smoking are about 2.67 times higher among those with throat cancer compared to those without throat cancer. --- ## **Final Answer for (a):** **The odds ratio of ever vs. never smoking, comparing study participants with and without throat cancer, is approximately 2.67.** --- Would you like to proceed to part (b)?

Certainly! Let's proceed step-by-step to solve all parts (a), (b), (c), (d), and (e) based on the data provided in the table: --- ### **Part (a): Calculate the Odds Ratio (OR) for ever smoking and throat cancer** **Step 1: Organize the data** | | Throat Cancer (Cases) | No Throat Cancer (Controls) | |------------------------|----------------------|-----------------------------| | Ever smoked | 40 | 20 | | Never smoked | 60 | 80 | --- **Step 2: Calculate the odds of exposure among cases and controls** - **Odds among cases:** \[ \frac{40}{60} = \frac{2}{3} \] - **Odds among controls:** \[ \frac{20}{80} = \frac{1}{4} \] --- **Step 3: Calculate the Odds Ratio (OR):** \[ \text{OR} = \frac{\frac{2}{3}}{\frac{1}{4}} = \frac{2}{3} \times 4 = \frac{8}{3} \approx 2.67 \] **_Final for (a):_** **Odds Ratio ≈ 2.67** — indicating that ever smokers have approximately 2.67 times higher odds of having throat cancer compared to never smokers. --- ### **Part (b): Calculate the Relative Risk (RR) of throat cancer for ever smokers** **Step 1: Calculate the incidence (risk) in each group** - **Risk among ever smokers:** \[ \frac{\text{Cases with smoking}}{\text{Total ever smokers}} = \frac{40}{40 + 20} = \frac{40}{60} = \frac{2}{3} \approx 0.6667 \] - **Risk among never smokers:** \[ \frac{60}{60 + 80} = \frac{60}{140} \approx 0.4286 \] --- **Step 2: Calculate the Relative Risk (RR):** \[ \text{RR} = \frac{\text{Risk among exposed}}{\text{Risk among unexposed}} = \frac{0.6667}{0.4286} \approx 1.56 \] **_Final for (b):_** **Relative Risk ≈ 1.56** — suggesting that ever smokers are about 1.56 times more likely to develop throat cancer than never smokers. --- ### **Part (c): Calculate the Attributable Risk (AR) in the exposed group** **Step 1: Use the formula:** \[ \text{AR} = \text{Risk in exposed} - \text{Risk in unexposed} \] - Risk in exposed (smokers): 0.6667 - Risk in unexposed (non-smokers): 0.4286 --- **Step 2: Calculate AR:** \[ 0.6667 - 0.4286 = 0.2381 \] **_Final for (c):_** **Attributable Risk ≈ 0.238** (or 23.8%) — meaning about 23.8% of throat cancer cases among smokers are attributable to smoking. --- ### **Part (d): Calculate the Population Attributable Risk (PAR)** **Step 1: Find the proportion of the total population that are exposed (smokers):** Total population = 100 (cases) + 100 (controls) = 200 Total smokers = 40 (cases) + 20 (controls) = 60 Proportion exposed in total population: \[ P_e = \frac{60}{200} = 0.3 \] --- **Step 2: Use the PAR formula:** \[ \text{PAR} = P_e \times (\text{RR} - 1) / [P_e \times (\text{RR} - 1) + 1] \] Where RR ≈ 1.56 \[ \text{PAR} = \frac{0.3 \times (1.56 - 1)}{0.3 \times (1.56 - 1) + 1} = \frac{0.3 \times 0.56}{0.3 \times 0.56 + 1} = \frac{0.168}{0.168 + 1} = \frac{0.168}{1.168} \approx 0.1438 \] **_Final for (d):_** **Population Attributable Risk ≈ 14.4%** — approximately 14.4% of throat cancer cases in the population could be prevented if smoking were eliminated. --- ### **Part (e): Interpret the findings** **Summary of results:** - **(a)** Odds Ratio ≈ 2.67 — smokers have higher odds of throat cancer. - **(b)** Relative Risk ≈ 1.56 — smokers are about 1.56 times more likely to develop throat cancer. - **(c)** Attributable Risk ≈ 23.8% — nearly a quarter of cases among smokers are attributable to smoking. - **(d)** Population Attributable Risk ≈ 14.4% — about 14.4% of cases in the population are due to smoking. **_Final interpretation:_** The data suggest that smoking significantly increases the risk of developing throat cancer. Approximately 23.8% of throat cancer cases among smokers are attributable to smoking, and eliminating smoking could potentially prevent around 14.4% of all cases in the population. --- **Would you like any further explanation or assistance?**

Certainly! Let's focus on part (e): **Interpret the findings** based on the previous calculations. ### **Part (e): Interpretation of the Findings** **Step 1: Summarize the key statistical measures** - **Odds Ratio (OR) ≈ 2.67:** Indicates that individuals who have ever smoked have approximately 2.67 times higher odds of developing throat cancer compared to those who never smoked. This suggests a strong association between smoking and throat cancer. - **Relative Risk (RR) ≈ 1.56:** Implies that ever smokers are about 1.56 times more likely to develop throat cancer than non-smokers. This reinforces the potential increased risk associated with smoking. - **Attributable Risk (AR) ≈ 23.8%:** Suggests that nearly a quarter (23.8%) of throat cancer cases among smokers can be attributed directly to smoking. If smoking were eliminated, these cases might potentially be prevented. - **Population Attributable Risk (PAR) ≈ 14.4%:** Indicates that approximately 14.4% of all throat cancer cases in the population could be prevented if smoking were eliminated, assuming a causal relationship. --- **Step 2: Overall interpretation** The data clearly show that smoking is associated with an increased risk of throat cancer. The high odds ratio and attributable risk highlight the public health importance of smoking cessation programs. Reducing smoking prevalence could significantly lower the incidence of throat cancer both among smokers and the general population. **In conclusion:** - **Smoking is a significant risk factor for throat cancer.** - **Approximately 24% of throat cancer cases among smokers are caused by smoking.** - **Public health efforts targeting smoking cessation could prevent a substantial proportion of throat cancer cases.** --- Would you like a more detailed explanation or any additional insights?