A manufacturing company claims that its new floodlight will last 1000 hours. Aft

Question

A manufacturing company claims that its new floodlight will last 1000 hours. After collecting a simple sample of size ten, you determine that a 90% confidence interval for the true mean number of hours that the floodlights will last, mu, is (959, 999). Which of the following are true? (Assume all tests mentioned below are two-sided) I) We have sufficient evidence to reject the null hypothesis that mu = 1000 at the 5% level. II) If an 80% confidence interval for the mean were determined here, the numerical value 958 would lie in this interval. III) If we were interested in testing the null hypothesis H_o: mu = 990, we know that the P-value must be > 10. A) More than one statement is true B) None are true C) Only I is true D) Only II is true E) Only III is true

Explanation / Answer

Answer to the question)

as the confidence level increases, the confidence interval widens

we got 90% confidence interval as 959 to 999

Thus if the confidence interval is 95% ( that is the significance level is 5%) then the interval will widen and it will include M = 1000. In that case we cannot reject the null that M = 1000 at 95% confidence level

Hence statement I is False

.

If the Confidence interval is 80% , then the limits will shrink further or get narrowed down. That is the lower limit at 90% is 959, will increase when the confidence level is 80%. Thus it will not include the value 958 for sure. Thus 958 will not lie in the interval .

Hence statement II is also not true

.

If we were interested in testing that M = 990 , then yes for sure M = 990 , lies in the interval 959 to 999. In that case we fail to reject the null hypothesis

Now this interval is obtained from 90% confidence limit , this implies significance level is 10% ( that is 100-90 = 10%)

Now since we fail to reject the null this is possible only when the p value > 0.10 ( or 10%) . Thus this statment makes perfect sense

[Decision rule: P value < significance value , reject the null ; P value > significance value , Fail to reject]

And hence statement III is true

Thus the correct answer choice is : ( E) Only Statment III is True

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