The mean water temperature downstream from a power plant coolingtower discharge pipe should be no more than 102oF. Pastexperienc
e has indicated that the standard deviation of temperatureis 2oF. The water temperature is measured on 9 randomlychosen days, and the average temperature is found to be100oF.(a) Is there evidence that the water temperature isacceptable atα = 0.05?(b) What is the P-value for thistest?(c) What is the probability of accepting thenull hypothesis at α = 0.05 if the water has atrue mean temperature of 106oF?
(a) Yes, there is evidence that the water temperature is acceptable at (b) 0.9987 (c) 6.647274e-06
Step-by-step explanation:
Let X be the random variable that represents the water temperature. The water temperature has been measured on n = 9 randomly chosen days (a small sample), the sample average temperature is = 100°F and = 2°F. We suppose that X is normally distributed.
We have the following null and alternative hypothesis
vs (upper-tail alternative)
We will use the test statistic
and the observed value is
.
(a) The rejection region is given by RR = {z | z > 1.6448} where 1.6448 is the 95th quantile of the standard normal distribution. Because the observed value -3 does not belong to RR, we fail to reject the null hypothesis. In other words, there is evidence that the water temperature is acceptable at .
(b) The p-value for this test is given by P(Z > -3) = 0.9987
(c) P(Accepting when ) = P(The observed value is not in RR when ) = P( < 1.6448 when ) = P( < 102 + (1.6448)() when ) = P( < 103.0965) when ) = P()) = P(Z < -4.3552) = 6.647274e-06
In order to solve this using only addition and subtraction, what we need to do is change the side each term is on. This will allow us to get the variable to be a positive number.