Question

For each coefficient estimate below, provide a confidence interval assuming that N=32 and d.f.=26.

(a) βˆ 1 = 1.4337; SE(βˆ 1) = 0.214. Provide a 90% confidence interval.

(b) βˆ 1 = −0.595; SE(βˆ 1) = 0.224. Provide a 95% confidence interval.

(c) βˆ 1 = 7.31; SE(βˆ 1) = 6.12. Provide a 90% confidence interval.

(d) βˆ 1 = −15.63; SE(βˆ 1) = 6.9. Provide a 95% confidence interval.

Answer 1

a)      The confidence interval for slope β₁

b₁ ± t_df,α/2 SE

Where b₁ = point estimate of slope

SE = standard error for slope b₁

t_26,0.0.05 = t critical value for two tailed with degrees of freedom 26 and 0.10 alpha level

t_26,0.05 = 1.706

b₁ ± t_26,0.05 SE = 1.4337 ± 1.706* (0.214) = (1.0686, 1.7988)

90% confidence interval for slope β₁ = (1.0686, 1.7988)

The 90% confidence interval is (1.0686, 1.7988).

b)      T_26,.0.025 = t critical value for two tailed with degrees of freedom 26 and 0.05 alpha level

t_26,0.025 = 2.056

b₁ ± t_26,0.05 SE = -0.595 ± 2.056* (0.224) = (-1.0555, -0.1345)

95% confidence interval for slope β₁ = (-1.0555, -0.1345)

The 95% confidence interval is (-1.0555, -0.1345).

c)      T_26,0.0.05 = t critical value for two tailed with degrees of freedom 26 and 0.10 alpha level

t_26,0.05 = 1.706

b₁ ± t_26,0.05 SE = 7.31 ± 1.706* (6.12) = (-3.1307, 17.7507)

90% confidence interval for slope β₁ = (-3.1307, 17.7507)

The 90% confidence interval is (-3.1307, 17.7507).

d)      T_26,.0.025 = t critical value for two tailed with degrees of freedom 26 and 0.05 alpha level

t_26,0.025 = 2.056

b₁ ± t_26,0.05 SE = -15.63 ± 2.056* (6.9) = (-29.8164, -1.4436)

95% confidence interval for slope β₁ = (-29.8164, -1.4436)

The 95% confidence interval is (-29.8164, -1.4436).