Given the following hypothesis testing situation: H 0 : = 50 H 1 : = 70 N = 25 s
ID: 3219388 • Letter: G
Question
Given the following hypothesis testing situation:
H0: = 50
H1: = 70
N = 25 sigma = 20 alpha = .01
a) Are any additional assumptions necessary to perform this test? If so, what?
b) Determine the critical value of X bar.
c) Draw a picture of the situation represented in this problem. Label the diagram as appropriately and as completely as possible. Be sure to indicate the critical value, the acceptance and rejection regions, and the hypothesized values of . Shade and label your diagram to show and .
d) If the observed (sample) mean is 35, what do you conclude (accept or reject null hypothesis)?
e) If the observed (sample) mean is 60, what do you conclude?
f) Compute (the probability of a type 2 error)
Explanation / Answer
a) yes there should be one more additional assumption, which is
the distribution should be normal of the population.
b) H0: mean= 50 H1: mean=70>50 [so alternative hypothesis is right tailed]
population standard deviation=sigma=20 alpha=level of significance=0.01 sample size=N=25
so H0 is rejected iff T=(Xbar-50)*sqrt(N)/sigma>tao0.01 where T~N(0,1) under H0 and tao0.01 is the upper 0.01 point of a N(0,1) distribution.
now tao0.01=2.326348 [using R]
so critical value of Xbar is
(Xbar-50)*sqrt(25)/20>2.326348
or, Xbar>50+2.326348*20/5=59.305392 [answer]
d) so if observed sample mean is 35 then it is below the critical value. hence the H0 is accepted and conclusion is mean=50
e) if observed sample mean is 60 then it is above critical value. hence H0 is rejected and conclusion is mean=70
f) P[type 2 error]=P[accepting H0 | H1 is true]
=P[Xbar<59.305392]
=P[(Xbar-70)*sqrt(25)/20<(59.305392-70)*sqrt(25)/20] [as under H1 Xbar follows normal with mean 70 and standard deviation 20/sqrt(25)]
=P[Z<-2.673652]=0.003751513 [answer]
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