Question

We are given that n=15, the sample mean Ῡ=2.5, the sample standard deviation s=1.5 and random variable Y distributed Normal with mean µ and variance σ², where both µ and σ² are unknown and we are being concentrated on testing the following set of hypothesis about the mean parameter of the population of interest.

We are to test:

H₀ : µ ≥ 3.0 versus H₁ : µ < 3.0.

Compute the following:

a) P- value of the test

b)    Probability of making Type II error and the power of this test at µ= 2.0

Answer 1

a)

Standard error , SE = s / = 1.5 / = 0.3872983

Test statistic, t = (Ῡ -  µ) / SE = (2.5 - 3) / 0.3872983 = -1.291

Degree of freedom = n-1 = 15-1 = 14

For left tail test, P-value = P(t < -1.291, df = 14) =  0.1088

b)

Assuming significance level of 0,05, Critical value of t for df = 14 is   -1.76

Critical value of sample mean Ῡ to reject null hypothesis is 3 - 1.76 * 0.3872983 =  2.318

Probability of making Type II error = P(Fail to reject H0 | H0 is False)

= P(Ῡ >  2.318 |  µ= 2.0)

= P[t > (2.318 - 2)/0.3872983]

= P[t >  0.8211] (df = 14)

= 0.2127

Power of the test = 1 - Probability of making Type II error

= 1 - 0.2127

= 0.7873