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AP Statistics 8.4 Expected Counts in Two-Way Tables- FRQs - Exam Style Questions

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Question

To compare success rates for treating allergies at two clinics that specialize in treating allergy sufferers, researchers selected random samples of patient records from the two clinics. The following table summarizes the data.
 Clinic AClinic BTotal
Unsuccessful treatment\( 51 \)\( 33 \)\( 84 \)
Successful treatment\( 88 \)\( 35 \)\( 123 \)
Total\( 139 \)\( 68 \)\( 207 \)
(a) (i) Complete the following table by recording the relative frequencies of successful and unsuccessful treatments at each clinic.
 Clinic AClinic B
Unsuccessful treatment  
Successful treatment  
(ii) Based on the relative frequency table in part (a-i), which clinic is more successful in treating allergy sufferers? Justify your answer.
(b) Based on the design of the study, would a statistically significant result allow the researchers to conclude that receiving treatments at the clinic you selected in part (a-i) causes a higher percentage of successful treatments than at the other clinic? Explain your answer.
A physician who worked at both clinics believed that it was important to separate the patients in the study by severity of the patient’s allergy (severe or mild). The physician constructed the following mosaic plot.
Based on the mosaic plot, the physician concluded the following:
For mild allergy sufferers, Clinic B was more successful in treating allergies.
For severe allergy sufferers, Clinic B was more successful in treating allergies.
(c) (i) For each clinic, which allergy severity is treated more successfully? Justify your answer.
• Clinic A:
• Clinic B:
(ii) For each clinic, which allergy severity is more likely to be treated? Justify your answer.
• Clinic A:
• Clinic B:
(d) Using your answers from part (c), give a reasonable explanation of why the more successful clinic identified in part (a-ii) is the same as or different from the physician’s conclusion that Clinic B is more successful in treating both severe and mild allergies.

Most-appropriate topic codes (CED):

• TOPIC 2.3: Statistics for Two Categorical Variables — parts (a), (c)
• TOPIC 3.1: Introducing Statistics: Do the Data We Collected Tell the Truth? — part (b)
• TOPIC 8.4: Expected Counts in Two-Way Tables — parts (c), (d)
▶️ Answer/Explanation
Detailed solution

(a)
(i) Relative frequency table:

 Clinic AClinic B
Unsuccessful treatment\( \frac{51}{139} \approx 0.367 \)\( \frac{33}{68} \approx 0.485 \)
Successful treatment\( \frac{88}{139} \approx 0.633 \)\( \frac{35}{68} \approx 0.515 \)

(ii) Clinic A is more successful. Clinic A has a higher success rate (\( 63.3\% \)) compared to Clinic B (\( 51.5\% \)).

(b)
No, a statistically significant result would not allow researchers to conclude causation. This is an observational study (not a randomized experiment) where patients were not randomly assigned to clinics. There may be confounding variables, such as allergy severity, that affect both the clinic a patient attends and the success of treatment.

(c)
(i)
• Clinic A: More successful in treating mild allergies (\( \frac{78}{104} = 0.75 \)) than severe (\( \frac{10}{35} \approx 0.286 \))
• Clinic B: More successful in treating mild allergies (\( \frac{11}{12} \approx 0.917 \)) than severe (\( \frac{24}{56} \approx 0.429 \))

(ii)
• Clinic A: More likely to treat mild allergy sufferers (\( \frac{104}{139} \approx 0.748 \)) than severe (\( \frac{35}{139} \approx 0.252 \))
• Clinic B: More likely to treat severe allergy sufferers (\( \frac{56}{68} \approx 0.824 \)) than mild (\( \frac{12}{68} \approx 0.176 \))

(d)
The overall success rate favored Clinic A in part (a-ii), which is different from the physician’s conclusion. This is an example of Simpson’s Paradox. Clinic A treats mostly mild cases (\( 74.8\% \)) which have higher success rates, while Clinic B treats mostly severe cases (\( 82.4\% \)) which have lower success rates. When we look within each severity level, Clinic B actually has higher success rates for both mild (\( 91.7\% \) vs \( 75.0\% \)) and severe (\( 42.9\% \) vs \( 28.6\% \)) allergies. The overall success rates are misleading because the clinics treat different types of patients.

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