Let X = tool life (measured in

  1. Let X = tool life (measured in some unit of time) of thecurrent tool type. Let Y = tool life of the new tool type beingconsidered. The standard deviations of X and Y are both equal to 2.Through tests the mean tool life of Y was shown to be statisticallysignificantly greater than X at a significance level of 0.05. Themean of X is estimated to be 10, and the mean of Y is estimated tobe 10.5. Assume that both X and Y are normally distributed.
    1. What is the probability that a randomly observed value of toollife X is greater than Y? (hint: the variance of the difference oftwo random variables is sum of their variances, and the differenceof two independent normal random variables is normallydistributed).
    2. How much greater would the mean of Y have to be so that thechance that a randomly observed value of tool life X is greaterthan Y is less than 0.05?


I answer the above questions using the basic knowledge ofstandard normal variable and their differences,

and the distribution of their differences.

The answers are given in subparts as given in the question.

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