Question

(a) Suppose you are given the following (x, y) data pairs 

x	1	3	6
y	2	1	7

Find the least-squares equation for these data (rounded to three digits after the decimal) 

(b) Now suppose you are given these (x, y) data pairs. 

x	2	1	7
y	1	3	6

Find the least-squares equation for these data (rounded to three digits after the decimal). 

(c) In the data for parts (a) and (b), did we simply exchange the x and y values of each data pair? 

   Yes 

   No 

(d) Solve your answer from part (a) for x (rounded to three digits after the decimal) 

  Do you get the least-squares equation of part (b) with the symbols x and y exchanged? 

   Yes

   No 

(e) In general, suppose we have the least-squares equation y=a+b x for a set of data pairs (x, y). If we solve this equation for x, will we necessarily get the least-squares equation for the set of data pairs (y, x), (with x and y exchanged)? Explain using parts (a) through (d).

• In general, switching x and y values produces the same least-squares equation.

• Switching x and y values sometimes produces the same least-squares equation and sometimes it is different.

• In general, switching x and y values produces a different least-squares equation.

Let x= boiler steam pressure in 100 lb/in² and let y= critical sheer strength of boilerplate steel joints in tons/in². We have the following data for a series of factory boilers.

x	4	5	6	8	10
y	3.4	4.2	6.3	10.9	13.3

(a) Make the logarithmic transformations x'=log(x) and y'=log (y). Then make a scatter plot of the (x', y') values. Does a linear equation seem to be a good fit to this plot?

• The transformed data does not fit a straight line well. The data seem to explode as x increases.

• The transformed data fit a straight line well.

• The transformed data does not fit a straight line well. The data seem to have a parabolic shape.

• The transformed data does not fit a straight line well. The data seem to explode as x decreases.

(b) Use the (x', y') data points and a calculator with regression keys to find the least-squares equation y'=a+bx'. What is the correlation coefficient? 

(c) Use the results of part (b) to find estimates for α and β in the power law y=αx^β. Write the power equation for the relationship between steam pressure and sheer strength of boiler plate steel. (Use 3 decimal places.)

Answer 1

Answer:

a) ŷ = 1.079X - 0.263

b) ŷ = 0.661X + 1.129

c) Yes

d) x= 0.927Y + 0.244

No, the equation is different from the answer we got in part(b)

e) In general, switching X and Y values produces a different result

for part (a)

y= 1.079X - 0.263

X= 0.927Y + 0.244

Here both the equations are different.

For part (b)

Y= 0.661X + 1.129

X= 1.512Y - 1.707

Here also Both the equations are different

Add per HOMEWORKLIB RULES, we can answer 1 question at a time.