The life in hours of a thermocouple used in a furnace is known to be approximate
ID: 3380919 • Letter: T
Question
The life in hours of a thermocouple used in a furnace is known to be approximately normally distributed, with a standared deviation of sigma=20 hours. A random sample of 15 thermocouples resulted in the following data: 553, 552, 567, 579, 550, 541, 537, 553, 552, 546, 538, 553, 581, 539, 529.
A. Is there evidence to support the claim that mean life exceeds 540 hours? Use a fixed test with alpha=0.05.
B. What is the P-value for this test?
C. Construct a 95% one-sided lower CI on the mean life.
D. Use the CI found in part (c) to test the hypothesis.
PLEASE SHOW ALL WORK! Thank you!
Explanation / Answer
a)
Formulating the null and alternative hypotheses,
Ho: u <= 540
Ha: u > 540
As we can see, this is a right tailed test.
Thus, getting the critical t,
df = n - 1 = 14
tcrit = + 1.761310136
Getting the test statistic, as
X = sample mean = 551.3333333
uo = hypothesized mean = 540
n = sample size = 15
s = standard deviation = 14.81151418
Thus, t = (X - uo) * sqrt(n) / s = 2.963492504
As t > 1.761, we REJECT THE NULL HYPOTHESIS.
There is sufficient evidence to support the claim that mean life exceeds 540 hours. [CONCLUSION]
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B)
Also, the p value is
p = 0.005133963 [answer]
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C)
Note that
Lower Bound = X - t(alpha) * s / sqrt(n)
where
alpha = (1 - confidence level) = 0.05
X = sample mean = 551.333333
t(alpha) = critical t for the confidence interval = 1.761310136
s = sample standard deviation = 14.811514
n = sample size = 15
df = n - 1 = 14
Thus,
Lower bound = 544.5975256
Thus, the confidence interval is u > 544.5975. [ANSWER]
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D)
As the whole confidence interval in c) is greater than 540, there is sufficient evidence to support the claim that mean life exceeds 540 hours. [CONCLUSION]
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