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Researchers want to test a new hand exercise plan for arthritis patients to see if they have improved flexibility. They recruited 110 patients with arthritis in their hands and randomly assigned them to one of two treatment groups: group 1 took their regular medicine and did the new hand exercises; and group 2 only took medicine. After one month, 42 of the 55 in group 1 improved in their flexibility, and 37 of the 55 in group 2 improved their flexibility.

Check the conditions for constructing a confidence interval for the difference in proportions.

- **Random condition**:
- **10% condition**:
- **Large Counts condition**:

Are all the conditions for inference met?

✔ Yes

Answer :

The conditions for constructing a confidence interval for the difference in proportions in the study are met. The random condition, 10% condition, and large counts condition are all satisfied. Therefore, a confidence interval can be constructed.

Random condition: The patients were randomly assigned to one of the two treatment groups. This condition is satisfied because randomness in assignment helps ensure that the groups are comparable.

10% condition: The sample size should be less than 10% of the population. With 110 patients, assuming the population of arthritis patients is much larger than 1,100, this condition is met.

Large Counts condition: Both groups need at least 10 successes and 10 failures. Group 1 has 42 successes (improved) and 13 failures (did not improve). Group 2 has 37 successes and 18 failures. Both numbers exceed 10, satisfying this condition.

Are all the conditions for inference met? ✔ Yes, all conditions for inference are satisfied.

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