The mean water temperature downstream from a discharge pipe at a power plant coo
ID: 3153921 • Letter: T
Question
The mean water temperature downstream from a discharge pipe at a power plant cooling tower should be no more than 100°F. Past experience has indicated that the standard deviation of temperature is 2°F. The water temperature is measured on nine randomly chosen days, and the average temperature is found to be 98°F.a) Is there evidence that the water temperature is acceptable at alpha = 0.05?
b) What is the P-value for this test?
c) What is the probability of accepting the null hypothesis at alpha = 0.05 if the water has a true mean temperature of 104°F? The mean water temperature downstream from a discharge pipe at a power plant cooling tower should be no more than 100°F. Past experience has indicated that the standard deviation of temperature is 2°F. The water temperature is measured on nine randomly chosen days, and the average temperature is found to be 98°F.
a) Is there evidence that the water temperature is acceptable at alpha = 0.05?
b) What is the P-value for this test?
c) What is the probability of accepting the null hypothesis at alpha = 0.05 if the water has a true mean temperature of 104°F?
a) Is there evidence that the water temperature is acceptable at alpha = 0.05?
b) What is the P-value for this test? a) Is there evidence that the water temperature is acceptable at alpha = 0.05?
b) What is the P-value for this test?
c) What is the probability of accepting the null hypothesis at alpha = 0.05 if the water has a true mean temperature of 104°F?
Explanation / Answer
The mean water temperature downstream from a discharge pipe at a power plant cooling tower should be no more than 100°F. Past experience has indicated that the standard deviation of temperature is 2°F. The water temperature is measured on nine randomly chosen days, and the average temperature is found to be 98°F.
Here, we have to use the one sample z test for the population mean. The null and alternative hypothesis is given as below:
Null hypothesis: H0: The mean water temperature downstream from a discharge pipe at a power plant cooling tower is 100°F.
Alternative hypothesis: Ha: The mean water temperature downstream from a discharge pipe at a power plant cooling tower is no more than 100°F.
H0: µ = 100 versus Ha: µ < 100
The test statistic formula is given as below:
Z = (xbar - µ) / [/sqrt(n)]
Here, we are given xbar = 98, µ = 100, = 2, sample size = n = 9
Z = (98 – 100) / [2/sqrt(9)]
Z = -3
Critical value for alpha 0.05 is given as -1.6449
Z Test of Hypothesis for the Mean
Data
Null Hypothesis m=
100
Level of Significance
0.05
Population Standard Deviation
2
Sample Size
9
Sample Mean
98
Intermediate Calculations
Standard Error of the Mean
0.6667
Z Test Statistic
-3.0000
Lower-Tail Test
Lower Critical Value
-1.6449
p-Value
0.0013
Reject the null hypothesis
a) Is there evidence that the water temperature is acceptable at alpha = 0.05?
We get the p-value as 0.0013 which is less than the given level of significance or alpha value 0.05, so we reject the null hypothesis that The mean water temperature downstream from a discharge pipe at a power plant cooling tower is 100°F. This means we conclude that the mean water temperature downstream from a discharge pipe at a power plant cooling tower is no more than 100°F. So, there is an evidence that the water temperature is acceptable at alpha = 0.05.
b) What is the P-value for this test?
The p-value is given as 0.0013.
c) What is the probability of accepting the null hypothesis at alpha = 0.05 if the water has a true mean temperature of 104°F?
If the sample is 104, then the test is given as below:
Here, we have to use the one sample z test for the population mean. The null and alternative hypothesis is given as below:
Null hypothesis: H0: The mean water temperature downstream from a discharge pipe at a power plant cooling tower is 104°F.
Alternative hypothesis: Ha: The mean water temperature downstream from a discharge pipe at a power plant cooling tower is no more than 104°F.
H0: µ = 104 versus Ha: µ < 104
The test statistic formula is given as below:
Z = (xbar - µ) / [/sqrt(n)]
Here, we are given xbar = 98, µ = 104, = 2, sample size = n = 9
Z = (98 – 104) / [2/sqrt(9)]
Z = -9
Critical value for alpha 0.05 is given as -1.6449
Z Test of Hypothesis for the Mean
Data
Null Hypothesis m=
104
Level of Significance
0.05
Population Standard Deviation
2
Sample Size
9
Sample Mean
98
Intermediate Calculations
Standard Error of the Mean
0.6667
Z Test Statistic
-9.0000
Lower-Tail Test
Lower Critical Value
-1.6449
p-Value
0.0000
Reject the null hypothesis
Z Test of Hypothesis for the Mean
Data
Null Hypothesis m=
100
Level of Significance
0.05
Population Standard Deviation
2
Sample Size
9
Sample Mean
98
Intermediate Calculations
Standard Error of the Mean
0.6667
Z Test Statistic
-3.0000
Lower-Tail Test
Lower Critical Value
-1.6449
p-Value
0.0013
Reject the null hypothesis
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