1.A manufacturer of battery-powered hand drills claims that, on average, their new drills can operate at an effective level for "at least 60 minutes" before it is necessary to re-charge the battery.Based on previous testing, the manufacturer assumes that the actual time, , before a battery requires re-charging is a normally distributed random variable with a standard deviation is minutes. A consumer group has asked you to test the validity of the manufacturer's claim that the population mean time for effective operation is at least 60 minutes.You are to use a random sample of 36 drills that are to be tested for the time that is required before re-charging is needed.You will use the sample mean time before re-charging is necessary as the basis for a hypothesis test to determine if there is significant evidence that the population mean time between re-charging is less than 60 minutes.The hypotheses appropriate for this test are: H_o: minutes and H_a: minutes.

(a)(4 points)You were instructed to use a level of significance of .First, determine the rejection region appropriate for this hypothesis test, and state a decision rule for rejecting the null hypothesis. Use 3 decimal places.

(b)(4 points)From your sample of 36 drills, you determine the sample mean time before re-charging is required to be 58.7 minutes. Using an English sentence, state the conclusion you should reach, in the context of this problem, based on your results in part (a) .

Be sure to indicate the reason upon which you are basing your conclusion.

(c)(6 points)A statistically-challenged analyst for the manufacturer tells you that "stronger evidence should be required before we conclude that the population mean time before re-charging is required is less than 60 minutes.After all, the sample mean time of 58.7 minutes was only 1.3 minutes less the our advertised time of a population mean of 60 minutes.We should repeat this test with a new sample and use the following criterion for rejecting the claim that the population mean time is equal to 60 minutes:Reject H_o: minutes if the sample mean time is less than 58 minutes."Note that this is a modified version of the rule you determined in part (a) using , and it re-defines the rejection region for the test.First, if this rule is used for rejecting the null hypothesis, what level of significance would be associated with the test? (You will need to use 5 decimal places for this part.) (HINT:). Second:Recalling that values for a level of significance are typically in the range from .005 to .10, does this approach to performing the test seem "reasonable"?

(d)(6 points)The manufacturer's analyst decides against the proposal in part (b) and instead suggests that the test should be performed based on a larger sample size of batteries to compute the sample mean time to re-charging.It is decided that you will repeat the test using a sample size of batteries.The sample mean time is found to be minutes.The analyst makes the following statement:"Compared to the sample mean in part (a), notice that the value for the sample mean computed with this sample size is closer to the value of population mean of 60 minutes given in the null hypothesis.As such, the strength of the evidence in favor of the claim that minutes is NOT AS STRONG as the evidence in the smaller sample in part (a)."Using the concept of the p-value associated with the observed value of the test statistics in parts (b) and (d) of this question, indicate whether or not you agree with the analyst's assessment.PROVIDE ALL RELEVANT COMPUTATIONS.

(e)(4 points)A level of significance of is to be used for the test described in part (d).The analyst states that "based on the results found in this sample of batteries, we have observed no evidence to support the claim that the claim that the population mean time to re-charging is less than 60 minutes."From a statistical standpoint, is this an appropriate description of the results of this hypothesis test?EXPLAIN WHY YOU FEEL THAT IT IS OR IS NOT.