10.60 Water specimens are taken from water used for cooling as it is being discharged from a power plant into a river. It has been determined that as long as the mean temperature of the discharged water is at most 1508F, there will be no negative effects on the river’s ecosystem. To investigate whether the plant is in compliance with regulations that prohibit a mean discharge water temperature above 1508F, a scientist will take 50 water specimens at randomly selected times and will record the water temperature of each specimen. She will then use a z statistic z 5 x 2 150 s

Ïn to decide between the hypotheses H0: m 5 150 and Ha: m . 150, where m is the mean temperature of discharged water. Assume that s is known to be 10.

a. Explain why use of the z statistic is appropriate in this setting.

b. Describe Type I and Type II errors in this context.

c. The rejection of H0 when z $ 1.8 corresponds to what value of a? (That is, what is the area under the z curve to the right of 1.8?)

d. Suppose that the actual value for m is 153 and that H0 is to be rejected if z $ 1.8. Draw a sketch

(similar to that of Figure 10.5) of the sampling distribution of x, and shade the region that would represent

b, the probability of making a Type II error.

e. For the hypotheses and test procedure described, compute the value of b when m 5 153. (Hint:

See Example 10.17.)

f. For the hypotheses and test procedure described, what is the value of b if m 5 160?

g. What would be the conclusion of the test if H0 is rejected when z $ 1.8 and x 5 152.4? What type of error might have been made in reaching this conclusion?