Question

# Suppose we observe Y1,...Yn from a normal distribution with unknown parameters such that ¯ Y =...

Suppose we observe Y1,...Yn from a normal distribution with unknown parameters such that ¯ Y = 24, s2 = 36, and n = 15.

(a) Find the rejection region of a level α = 0.05 test of H0 : µ = 20 vs. H1 : µ 6= 20. Would this test reject with the given data?

(b) Find the rejection region of a level α = 0.05 test of H0 : µ ≤ 20 vs. H1 : µ > 20. Would this test reject with the given data?

(c) Will the p-value for the given data be smaller in part (a) or (b)? Why?

x̅ = 24, s = 6, n = 15

a) Null and Alternative hypothesis:

Ho : µ = 20 ; H1 : µ ≠ 20

Test statistic:

t = (x̅ - µ)/(s/√n) = (24 - 20)/(6/√15) = 2.5820

df = n-1 = 14

Critical value :

Two tailed critical value, t-crit = T.INV.2T(0.05, 14) = 2.145

Rejection region:

Reject Ho if t < -2.145 or if t > 2.145

Decision:

As t = 2.5820 > 2.145, Reject the null hypothesis.

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b)

Null and Alternative hypothesis:

Ho : µ ≤ 20 ; H1 : µ > 20

Critical value :

Right tailed critical value, t-crit = ABS(T.INV(0.05, 14)) = 1.761

Rejection region:

Reject Ho if t > 1.761

Decision:

As t = 2.5820 > 1.761, Reject the null hypothesis

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c) The p-value will be smaller in part b).

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