Question

Find the 95% confidence interval for the slope in each of the following settings. (Round your answers to three decimal places.)

a) n = 20, ŷ = 27.5 + 1.8x, and SE_b1 = 0.85

b) n = 30, ŷ = 30.6 + 2.1x, and SE_b1 = 1.05

c) n = 100, ŷ = 29.5 + 2.1x, and SE_b1 = 1.05

Answer 1

Here, we need to find the 95% confidence interval for the slope

we know that the least square regression line is as follows,

where, = dependent outcome variable and x is the independent variable

is the intercept and   is the slope parameter

(A)

we have given that

n= 20

i.e. =intercept=27.5 and   =slope=1.8

=Standard error of slope = 0.85

Now we cab find the 95% confidence interval for the slope  

The formula is as follows

where  b1 is the slope

first, we can find the critical value

Degrees of freedom = n-2 = 20 -2 =18

c=confidnece level =0.95

= level of significance= 1-c =1-0.95=0.05

this 95% CI is two tailed

t-critical = 2.101 Using Excel software =TINV(probablity =0.05, D.F=18)

Now,

The 95% CI is (0.014, 1.814)

(B)

we have given that

n= 30

i.e. =intercept=30.6 and   =slope=2.1

=Standard error of slope = 1.05

Now we cab find the 95% confidence interval for the slope  

The formula is as follows

where  b1 is the slope

first, we can find the critical value

Degrees of freedom = n-2 = 30 -2 =28

c=confidnece level =0.95

= level of significance= 1-c =1-0.95=0.05

this 95% CI is two tailed

t-critical =2.048   Using Excel software =TINV(probablity =0.05, D.F=28)

Now,

The 95% CI is (-0.050, 4.250)

(C)

we have given that

n= 100

i.e. =intercept=29.5 and   =slope=2.1

=Standard error of slope = 1.05

Now we cab find the 95% confidence interval for the slope  

The formula is as follows

where  b1 is the slope

first, we can find the critical value

Degrees of freedom = n-2 = 100 -2 =98

c=confidnece level =0.95

= level of significance= 1-c =1-0.95=0.05

this 95% CI is two tailed

t-critical = 1.984 Using Excel software =TINV(probablity =0.05, D.F=98)

Now,

The 95% CI is (0.016, 4.184)