Homework SourseThe life in hours of a thermocouple used in a furnace is kno
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!
Solution
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]
