Question

1. as of 2018 the world population is 7.616 billion people and is growing at a rate of 1.1% per year.

(a) write and equation to model the population growth, where p(t) is population in billions of people and t is time in years.

(b) what is the predicted population for 2050?

1. a simple random sample of 50 high school students is used to collect data on gpa and sat mathematics scores. suppose the scatter plot of data is liner and produces a regression equation of predicted sat=550.8+56.3(gpa).

(a) interpret the y-intercept of the equation. does it make sense in this context?

(b) interpret the slope of the equation.

(c) rewrite the equation by solving for gpa.

(d) suppose someone uses this equation to predict a score of 660 on the mathematics portion of the sat. use your equation above to find their gpa.

Explanation:

Problem 2 (a)

Step1: Identify Growth Model

Population growth is exponential, formula: \( P(t) = P_0(1 + r)^t \), where \( P_0 \) is initial population, \( r \) is growth rate.
Initial population \( P_0 = 7.616 \) billion, growth rate \( r = 0.0114 \) (1.14% converted to decimal).

Step2: Write the Equation

Substitute \( P_0 \) and \( r \) into the formula: \( P(t) = 7.616(1 + 0.0114)^t = 7.616(1.0114)^t \).

Step1: Calculate Time \( t \)

From 2018 to 2050, \( t = 2050 - 2018 = 32 \) years.

Step2: Substitute \( t \) into Equation

Use \( P(t) = 7.616(1.0114)^{32} \). Calculate \( (1.0114)^{32} \approx 1.428 \) (using exponentiation). Then \( P(32) \approx 7.616 \times 1.428 \approx 10.88 \) billion.

The regression equation is \( \text{Predicted SAT} = 550.8 + 56.3(\text{GPA}) \). The y - intercept is 550.8, which represents the predicted SAT score when GPA = 0. In context, a GPA of 0 means no academic achievement, but SAT scores start above 0, so it may not make sense (as real students have GPA > 0, and SAT scores for GPA = 0 are unrealistic).

Answer:

\( P(t) = 7.616(1.0114)^t \)

Problem 2 (b)