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. $$ \hat{p_{T}}-\hat{p_{N}}= $$

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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:**
$$
\hat{p_T} - \hat{p_N} = 0.550
$$
The difference in proportions is 0.550.

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