Question 1192375: 42 Marijuana Usage among High School Students Figure 2.14 illustrates some sur vey data regarding the usage of marijuana among high school students. A sample of high werpen (ca school students was taken every 2 years between 1979 and 1989. The data in Figure 2.14 what reflects the percentage of students surveyed who indicated they had used marijuana during the previous 30 days. The data points suggest that the percentage of students having used marijuana is decreasing at an approximately linear rate over time. If the data points for 1979 (1, 36.5) and 1989 (11, 16.5) are used to estimate the linear equation which relates the percentage of students P to time t (t = 1 corresponding to 1979): (a) Determine the slope-intercept form of the estimating equation. (b) Forecast the expected percentage for 1991 and 1995. (c) Interpret the meaning of the slope and P intercept. word problem
Answer by CPhill(1959)   (Show Source):
You can put this solution on YOUR website!
**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.
|
|
|
