Question 1194224
**a. Construct a 95% Confidence Interval for the Average Time All Men Spend Doing Household Chores**

* **Given:**
    * Sample Mean (x̄) = 54 minutes
    * Sample Standard Deviation (s) = 12.7 minutes
    * Sample Size (n) = 1136 men
    * Confidence Level = 95% 

* **Find the Critical Value (zα/2):**
    * For a 95% confidence level, α = 0.05 
    * zα/2 = 1.96 (from the standard normal distribution table)

* **Calculate the Standard Error of the Mean (SEM):**
    * SEM = s / √n 
    * SEM = 12.7 / √1136 
    * SEM ≈ 0.377

* **Calculate the Margin of Error (ME):**
    * ME = zα/2 * SEM 
    * ME = 1.96 * 0.377 
    * ME ≈ 0.739

* **Construct the Confidence Interval:**
    * Lower Limit = x̄ - ME = 54 - 0.739 = 53.261 minutes
    * Upper Limit = x̄ + ME = 54 + 0.739 = 54.739 minutes

* **95% Confidence Interval for All Men:** (53.261 minutes, 54.739 minutes)

**b. Construct a 95% Confidence Interval for the Average Time Women Spend Doing Household Chores**

* **Given:**
    * Sample Mean (x̄) = 72 minutes
    * Sample Standard Deviation (s) = 10.4 minutes
    * Sample Size (n) = 795 women
    * Confidence Level = 95% 

* **Find the Critical Value (zα/2):**
    * Same as part (a): zα/2 = 1.96

* **Calculate the Standard Error of the Mean (SEM):**
    * SEM = s / √n 
    * SEM = 10.4 / √795 
    * SEM ≈ 0.368

* **Calculate the Margin of Error (ME):**
    * ME = zα/2 * SEM 
    * ME = 1.96 * 0.368 
    * ME ≈ 0.721

* **Construct the Confidence Interval:**
    * Lower Limit = x̄ - ME = 72 - 0.721 = 71.279 minutes
    * Upper Limit = x̄ + ME = 72 + 0.721 = 72.721 minutes

* **95% Confidence Interval for All Women:** (71.279 minutes, 72.721 minutes)

**Interpretation:**

* We are 95% confident that the true average time all men spend on household chores lies between 53.261 and 54.739 minutes.
* We are 95% confident that the true average time all women spend on household chores lies between 71.279 and 72.721 minutes.