0.425 (c) Of the people recelving the therapy session, what proportion reported sleep improvements? Round your answer three decimal places. i 0.700 (d) What proportion of people who did not receive therapy reported sleep improvements? Round your answer three decimal places. i 0.150 (e) If we use \( \hat{p_{T}} \) to denote the proportion from part (c) and use \( \hat{p_{N}} \) to denote the proportion from part (d), calculate the difference in proportion reporting sleep improvements, \( \hat{p_{T}}-\hat{p_{N}} \), between those getting therapy and those not getting therapy. Round your answer three decimal places. \[ \hat{p_{T}}-\hat{p_{N}}= \]

To calculate the difference in proportions reporting sleep improvements between those receiving therapy and those not receiving therapy, follow these steps: 1. **Proportion of Therapy Group (\( \hat{p_T} \)):** From part (c), the proportion of people receiving therapy who reported sleep improvements is: \[ \hat{p_T} = 0.700 \] 2. **Proportion of Non-Therapy Group (\( \hat{p_N} \)):** From part (d), the proportion of people not receiving therapy who reported sleep improvements is: \[ \hat{p_N} = 0.150 \] 3. **Difference in Proportions (\( \hat{p_T} - \hat{p_N} \)):** Subtract the proportion of the non-therapy group from the therapy group: \[ \hat{p_T} - \hat{p_N} = 0.700 - 0.150 = 0.550 \] **Final Answer:** \[ \hat{p_T} - \hat{p_N} = 0.550 \]