Question 1192375

**1. Find the Slope**

* **Slope (m):** (Change in P) / (Change in t) = (16.5 - 36.5) / (11 - 1) = -20 / 10 = -2

**2. Find the Y-intercept (b)**

* **Use the point-slope form:** P - P1 = m(t - t1) 
    * Use the point (1, 36.5) and the slope (-2)
    * P - 36.5 = -2(t - 1)
    * P - 36.5 = -2t + 2
    * P = -2t + 38.5

**a) Slope-Intercept Form of the Estimating Equation:**

* **P = -2t + 38.5** 
    * where:
        * P is the percentage of students using marijuana
        * t is the time in years since 1978 (t = 1 corresponds to 1979)

**b) Forecast the Expected Percentage**

* **1991:** 
    * t = 1991 - 1978 = 13
    * P = -2(13) + 38.5 = 12.5%

* **1995:** 
    * t = 1995 - 1978 = 17
    * P = -2(17) + 38.5 = 4.5%

**c) Interpretation**

* **Slope:** The slope of -2 indicates that the percentage of students using marijuana is estimated to decrease by 2% per year.
* **P-intercept (38.5):** This represents the estimated percentage of students using marijuana in 1978 (when t = 0). However, since the model is based on data from 1979 onwards, this intercept may not have a practical interpretation in this specific context.

**Note:**

* This linear model provides an estimation based on the given data. 
* Actual trends may not perfectly follow this linear pattern. 
* Other factors could influence marijuana usage among high school students, and this model may not capture all of these factors.
**a) Determine the slope-intercept form of the estimating equation.**

* **Given Data Points:**
    * (t1, P1) = (1, 36.5) 
    * (t2, P2) = (11, 16.5) 

* **Calculate Slope (m):**
    * m = (P2 - P1) / (t2 - t1) 
    * m = (16.5 - 36.5) / (11 - 1) 
    * m = -20 / 10 
    * m = -2

* **Calculate y-intercept (b):**
    * Use the point-slope form: P - P1 = m(t - t1)
    * P - 36.5 = -2(t - 1)
    * P - 36.5 = -2t + 2
    * P = -2t + 38.5

* **Slope-intercept form:** 
    * **P = -2t + 38.5** 

**b) Forecast the expected percentage for 1991 and 1995.**

* **For 1991 (t = 13):**
    * P = -2(13) + 38.5 
    * P = -26 + 38.5 
    * P = 12.5%

* **For 1995 (t = 17):**
    * P = -2(17) + 38.5 
    * P = -34 + 38.5 
    * P = 4.5%

**c) Interpret the meaning of the slope and P intercept.**

* **Slope (-2):** 
    * The slope represents the rate of change in the percentage of students using marijuana per year. 
    * In this case, it indicates that the percentage of students using marijuana is decreasing by 2% per year.

* **P-intercept (38.5):**
    * The P-intercept represents the estimated percentage of students using marijuana in the base year (1979, when t = 1). 
    * It suggests that in 1979, approximately 38.5% of high school students used marijuana.

**Note:** 
* This model is based on a linear assumption and may not accurately predict future trends. 
* Factors other than time can influence marijuana usage among high school students.

I hope this helps! Let me know if you have any further questions.