Question

An experiment was performed on a certain metal to determine if the strength is a function of heating time (hours). Results based on 25 metal sheets are given below. Use the simple linear regression model.

∑X = 50

∑X2 = 200

∑Y = 75

∑Y2 = 1600

∑XY = 400

Find the estimated y intercept and slope. Write the equation of the least squares regression line and explain the coefficients. Estimate Y when X is equal to 4 hours. Also determine the standard error, the Mean Square Error, the coefficient of determination and the coefficient of correlation. Check the relation between correlation coefficient and Coefficient of Determination. Test the significance of the slope.

Please show all work, please type out so it is legible, thank you

Answer 1

Answer:

Given,

Ʃ x = 50

Ʃ x² = 200

Ʃ y = 75

Ʃ y² = 1600

Ʃ xy = 400

Sample size, n = 25

Now let us consider,

x̅ = Ʃx/n

substitute values

= 50/25

x̅ = 2

y̅ = Ʃy/n

substitute values

= 75/25

y̅ = 3

Now,

SSxx = Ʃx² - (Ʃx)²/n

substitute values in above formula

= 200 - (50)²/25

= 200 - 100

SSxx = 100

SSyy = Ʃy² - (Ʃy)²/n

substitute values in above formula

= 1600 - (75)²/25

= 1600 - 225

SSyy = 1375

SSxy = Ʃxy - (Ʃx)(Ʃy)/n

substitute values in above formula to get required value

= 400 - (50)(75)/25

= 400 - 150

SSxy = 250

Slope, b = SSxy/SSxx

substitute values to get slope

= 250/100

b = 2.5

consider,

y-intercept,

a = y̅ - b* x̅

= 3 - (2.5)*2

= 3 - 5

a = - 2

Now we have to consider the regression equation :

ŷ = - 2 + (2.5)*x

substitute x = 4 in above formula

So the predicted value of y is given below

ŷ = - 2 + (2.5) * 4

= - 2 + 10

ŷ = 8

SSE = SSyy - SSxy²/SSxx

substitute the known values

= 1375 - (250)²/100

= 1375 - 625

SSE = 750

Standard error = √(SSE/(n-2))

= √(750/(25 - 2))

= √32.61

Standard error = 5.71040

Mean square error

MSE = SSE/(n-2)

= 750/(25-2)

= 750/23

MSE = 32.6087

Correlation coefficient, r = SSxy/√(SSxx*SSyy)

= 250/√(100*1375)

= 250/370.81

r = 0.6742

Coefficient of determination,

r² = (SSxy)²/(SSxx*SSyy)

= (250)²/(100*1375)

= 62500/137500

r² = 0.4545

Correlation coefficient is given as the square root of coefficient of determination

Slope Hypothesis test: can be given as follows

Null hypothesis

Ho: β₁ = 0

Alternative hypothesis

Ha: β₁ ≠ 0

Consider test statistic:

t = b / (SE/√SSxx)

now after substituting the values and then we get

t = 4.3780

degree of freedom = n - 2

= 25 - 2

df = 23

Critical value,

Corresponding t value = t(0.05, 23)

t = 2.067

P value = corresponding p value for t(4.378), 23 is 0.0002

So

P value = 0.0002

Here we observe that p-value < alpha , so we reject the null hypothesis Ho.