Question

1. ] X~NORM(44.0, 3.2). A sample mean of 46.7 is taken, for n=30. Given H₀: m=44.0, H₁: m > 44.0, and a=0.05. Note this is a test on the mean for a single population.
   1. What test statistic should be used? (Z, T, c², F?)? Why?
   2. What is the critical value of the test statistic to use for this test? What is the accept region (accept H₀) and reject region (reject H₀)?
   3. Based on the critical value, what is your conclusion regarding this test given the sample mean above?
   4. What impact would increasing a (say to 0.1) have on the critical value? On the test? What does a control?
   5. What is the one-sided p-value for the sample mean above?
   6. Based on the p-value from part (e), what is your conclusion regarding this test given the sample mean above? Is it different than from part (c)
   7. Suppose you are instead doing a two-sided test: Given H₀: m=44.0, H₁: m <> 44.0 and a=0.05. Calculate the test critical values and state the conclusion of the test for the sample mean above.
   8. Supposed that, unknown to you, the distribution of X had shifted to ~NORM(43.8, 3.3). For n=30, calculate power and b (type II error) for the two-sided hypothesis test of part (g).

Answer 1

a.

Z test statistic should be used, because we know the population standard deviation.

b.

Critical value = Z_0.05 = 1.64

The acceptance region is the region below z = 1.64, and rejection region is region above z = 1.64.

c.

Test statistic,

Since calculated value is greater than critical value, we reject null hypothesis and conclude that mean is greater than 44.

d.

If we change a = 0.05 to a =0.1 then critical value = 1.28, rejection region get wider as compared at a =0.05

e.

P-value = P( z > 4.62) = 1 - P(z< 4.62) = 1 - 0.999998 = 0.000002

f.

Since p-value is less than a = 0.05, hence we reject null hypothesis.

This result is same as result in part c.