Question

Please help with this statistics problem.

Exercise 2: In a study, scientists examine data on the "Sea temperature" (variable r, in celsius) and "Coral growth" (response y, in millimetres) over a several-years period in the Gulf of Mexico and the Caribbeans. The scatterplot in graph A displays the relationship between and y 262 26.7 263 264 26.5 26.0 Sea Tamperature Answer the following questions providing all the mathematical details. 1. Which regression line (in graphs A and B) fits well the scatterplot? (A) (B) 26.1 26 2 26 7 26.3 2 5 283 24 2.5 27 Temeralre Sea Taroesture 2. In graph A, find the intercept a and slope b of the regression line (explain your answer) 3. Interpret the slope b in the context of the problem. 4. Predict the value of Growth from a Temperature = 30. 5. Suppose = 26.48 and s = 0.21 are the mean and standard-deviation of a, and r = 0.81 is the correlation. What are the values of ỹ and s? (explain mathematically your answer) 6. Does this regression method accurately predict well the Growth? Explain.

Answer 1

1. Graph A fits well the scatterplot

2. slope = change in y / change in x

Let us consider two points (26.1 , 9.2) and (26.3, 8.9)

slope, b = (8.9-9.2)/(26.3-26.1) = -1.5

y- intercept(a) is the point where the line meets the y axis

We know that y = a+bx

For a point (26.1 , 9.2)

9.2= a - 1.5*26.1

Solving , a= 48.35

3.Slope =-1.5

For each unit increase in x variable , y variable decreases by 1.5

4. the line of regression is

y=a+bx

growth = 48.35- 1.5*temperature

temp=30

growth = 48.35 - 30*1.5 = 3.35

5.We know that

For y =a +bx

b= r *(sy/sx)

using the values , b = -1.5 , r= -0.81 ,sx= 0.21

sy = 0.39

Also ,

a y-br

Thus ,

= 8.63