Question

The heat evolved in calories per gram of a cement mixture is approximately normally distributed. The population mean is thought to be 100, and the population standard deviation σ is 2. You wish to test H0 : µ = 100 versus H1 : µ 6= 100. Note that this is a two-sided test and they give you σ, the population standard deviation. (a) State the distribution of X¯ assuming that the null is true and n = 9.

(b) Find the boundary of the rejection region for the test statistic (these critical values will be z-values) if the type I error probability is α = 0.01.

(c) Find the boundary of the rejection region in terms of ¯x if the type I error probability is α = 0.01. In other words, how much lower than 100 must X¯ be to reject and how much higher than 100 must X¯ be to reject. You will have an ¯xlow and an ¯xhigh defining the rejection region. HINT: You are un-standardizing your z from part (b) here.

(d) What is the type I error probability α for the test if the acceptance region for the hypothesis test is instead defined as 98.5 ≤ x¯ ≤ 101.5? Recall that α is the probability of rejecting H0 when H0 is actually true.

Answer 1

a)

Xbar follow normal distribution with mean = mu = 100

and sd = sigma/sqrt(n) = 2/sqrt(9) = 2/3

b)

we reject the null if Z > 2.576 or less than -2.576

c)

X = mean + z*sd

lower = 100 -2.576*2/3 = 98.2827

upper = 100 + 2.576*2/3 = 101.7173

d)

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