Question

Given a variable that has a t distribution with the specified degrees of freedom, what percentage of the time will its value fall in the indicated region? (Round your answers to one decimal place.)

(a) 10 df, between -2.23 and 2.23
___ %

(b) 10 df, between -3.17 and 3.17
___%

(c) 24 df, between -2.49 and 2.49
___%

(d) 24 df, between -3.47 and 3.47
___%

(e) 23 df, outside the interval from -2.81 to 2.81
___%

(f) 24 df, to the right of 2.80
___ %

(g) 10 df, to the left of -1.81
___ %

Answer 1

a) P(-2.23 < t₁₀ < 2.23) = P(t₁₀ < 2.23) - P(t₁₀ < -2.23) = 0.9751 - 0.0249 = 0.9502 = 95.02%

b) P(-3.17 < t₁₀ < 3.17) = P(t₁₀ < 3.17) - P(t₁₀ < -3.17) = 0.995 - 0.005 = 0.99 = 99%

c) P(-2.49 < t₂₄ < 2.49) = P(t₂₄ < 2.49) - P(t₂₄ < -2.49) = 0.99 - 0.01 = 0.98 = 98%

d) P(-3.47 < t₂₄ < 3.47) = P(t₂₄ < 3.47) - P(t₂₄ < -3.47) = 0.999 - 0.001 = 0.998 = 99.8%

e) 1 - P(-2.81 < t₂₃ < 2.81) = 1 - (P(t₂₃ < 2.81) - P(t₂₃ < -2.81)) = 1 - (0.995 - 0.005) = 1 - 0.99 = 0.01 = 1%

f) P(t₂₄ > 2.8) = 1 - P(t₂₄ < 2.8) = 1 - 0.995 = 0.005 = 0.5%

g) P(t₁₀ < -1.81) = 0.0502 = 5.02%