Radon is a radioactive gas that occurs naturally underground. It finds its way i
ID: 3171383 • Letter: R
Question
Radon is a radioactive gas that occurs naturally underground. It finds its way into the water supply and is emitted through shower heads. Inspectors for the Environmental Protection Agency wonder whether the orifice diameters of shower heads have any effect on the amount of radon emitted. They decide to test orifice diameters of 0.37mm, 0.51mm, 0.71mm, and 1.02mm. Their hypotheses are H0:0.37=0.51=0.71=1.02{"version":"1.1","math":"H_0: mu_{0.37} = mu_{0.51} = mu_{0.71} = mu_{1.02}"} and Ha:At least one orifice diameter emits a different amount of radon than shower heads of the other diameters{"version":"1.1","math":"H_a: ext{At least one orifice diameter emits a different amount of radon than shower heads of the other diameters}"}. They choose =0.01{"version":"1.1","math":"lpha = 0.01"} and randomly select 4 shower heads of each orifice diameter. Assume the requirements for their hypothesis test are satisfied.
Input the test statistic.
Input the P-value.
Regardless of your answer to the previous question, suppose the inspectors of the previous question failed to reject H0{"version":"1.1","math":"H_0"}. Which of the following statements best expresses the conclusion of their test?
There is sufficient evidence to conclude that orifices of different sizes emit the same amount of radon gas.
There is insufficient evidence to conclude that orifices of different sizes emit the same amount of radon gas.
There is sufficient evidence to conclude that at least one of the orifices of different sizes emit different amounts of radon gas.
There is insufficient evidence to conclude that at least one of orifices of different sizes emit different amounts of radon gas.
Diameter Radon Released 0.37 80 0.37 83 0.37 83 0.37 85 0.51 75 0.51 75 0.51 79 0.51 79 0.71 74 0.71 73 0.71 76 0.71 77 1.02 67 1.02 72 1.02 74 1.02 74Explanation / Answer
Solution:
Here, we have to check whether the average radon released is same for the four different diameter or not. For checking this hypothesis or claim we have to use the one way analysis of variance or ANOVA test. The ANOVA table by using excel is given as below:
Diameter 0.37
Diameter 0.51
Diameter 0.71
Diameter 1.02
80
75
74
67
83
75
73
72
83
79
76
74
85
79
77
74
ANOVA: Single Factor
SUMMARY
Groups
Count
Sum
Average
Variance
Diameter 0.37
4
331
82.75
4.25
Diameter 0.51
4
308
77
5.333333
Diameter 0.71
4
300
75
3.333333
Diameter 1.02
4
287
71.75
10.91667
ANOVA
Source of Variation
SS
df
MS
F
P-value
F crit
Between Groups
256.25
3
85.41667
14.33566
0.000285
3.490295
Within Groups
71.5
12
5.958333
Total
327.75
15
For this ANOVA the test statistic value F is given as 14.3357.
The P-value for this ANOVA test is given as 0.000285 which is less than the level of significance or alpha value 0.01. So, we reject the null hypothesis that the average radon released is same for the four different diameters. This means there is sufficient evidence to conclude that at least one of the orifices of different sizes emit different amounts of radon gas.
Correct Answer: C. There is sufficient evidence to conclude that at least one of the orifices of different sizes emit different amounts of radon gas.
Diameter 0.37
Diameter 0.51
Diameter 0.71
Diameter 1.02
80
75
74
67
83
75
73
72
83
79
76
74
85
79
77
74
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