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I answered this question in three ways: classical, p-value, and confidence interval.

Data:553, 552, 567, 579, 550, 541, 537, 553, 552, 546, 538, 553, 581, 539, 529

n = 15 sample mean = 551.3 normally distributed, sigma = 20

Question: Is there evidence to support the claim that the mean life exceeds 540 hours?

Classical: right-tailed CV = 1.645 z = 2.19; Since 2.19 > 1.645, reject Ho

p-value: z = 2.19, A = 0.4857, Since p-value = 0.0143 < 0.05, reject Ho

Confidence interval: mean <= 551.3 + 1.645 * 20 div sqrt 15 equals 558? Since 540 is <= 558, do not reject Ho. Am i right? Thanks for the help

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If your confidence interval does not CROSS the target value assigned in Ho then there is no evidence to support Ho and you should reject it.

Only if lowerCIHo do you not reject Ho. If both CIs lie on the same side of Ho you reject it.

In your case the mean is above the target value of 540 and so it is the lower confidence interval that will matter in this case. If it is greater than Ho then reject it, if it is less do not.

Typically you would calculate standard deviation from your data rather than use a population reference value (e.g. from a textbook) since you want to address Ho regarding your data specifically as the value of the standard deviation calculated in a batch will capture batch specific biases and effects so may differ from the overall population based Sigma. The standard deviation of the data presented is around 14 rather than 20.

CIs and p-values are testing the hypothesis making the assumption of equivalent sampling from the overall population, so using population derived values will not give the correct results as these do not capture the variation characteristics of the sampling method.

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  • $\begingroup$ The sigma was given in the book. The sample standard deviation is around 14. $\endgroup$ – user196863 Mar 2 '18 at 5:03
  • $\begingroup$ redo your calculations using the data standard deviation and then check if the CI crosses the Ho value. I would be surprised if they did. $\endgroup$ – ReneBt Mar 2 '18 at 15:58

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