The mean water downstream from a power plant cooling tower discharge pipe should

Question

The mean water downstream from a power plant cooling tower discharge pipe should be no more than 100F. It is assumed that the water temperature follows a normal distribution with parameter µ and . An environmental engineer is interested in the following hypothesis.

H0 : µ =100 F

H1 : µ > 100F

Part A:

Past experience indicates that the standard deviation of temperature is 4 F. The water temperature is measured on 7 randomly chosen days and the average temperature found is found to 102.1 F.

Should the water temperature be judged acceptable with type 1 error alpha = 0.01?

What is the P-value for this test?

What is the probability of accepting the null hypothesis at alpha = 0.01 if the true water temperature has a mean of 108 F.

What sample size would be required to detect a true mean of 102 F if we wanted the power of the test to be at least 0.95

Explanation / Answer

Since the sample size is small, we use a t-table to find the critical value

test statistic = (102.1-100)/(4/sqrt(7)) = 1.389

From a t-table with 6 degrees of freedom, the critical value for a one-sided alpha of 0.01 is 3.143

Since the test statistic < critical value, we can accept the null hypothesis that the true mean of the water is 100 F

P-value = probability corresponding to the test statistic of 1.389 = 0.107097

If true water temperature has a mean of 108, test statistic = (102.1-108/(4/sqrt(7)) = -3.9024

since the test statistic is higher than the critical value 3.143, the results are significant and the null hypothesis can be rejected. p-value = .003992, is the probability of accepting the null hypothesis.

Samplemean = 102.1, true mean = 102, sample size = n; critical value corresponding to power of 0.95 = 1.943

test statistic = (102.1-102)/(4/sqrt(n)) =1.943

n = (1.943*4/0.1)^2 = 6040.38, rounded up to 6041

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