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# Two-Sample Tests of Hypotheses

Suppose you were the manufacturer of a new and improved light bulb that you claim has an average service life more than twice the life of your nearest competitor's product. You want to test your product to be sure. You test 500 of your bulbs in the lab and find that the average life of your bulbs is 2021 hours. Your competitor claims an average service life of 1000 hours. What hypotheses would you test? State the null and alternative hypotheses both verbally and algebraically

Our light bulbs have an average service life more than twice that of our competitor's product.

Our competitor's product has an average service life of 1000 hours, so the alternative hypothesis becomes this:

Our light bulbs have an average service life of more than 2000 hours.

Or algebraically:

mu > 2000

1st part

Based on this scenario I need to discuss the same scenario, but from the perspective of having two samples: 500 of our light bulbs and 500 of the competition's bulbs. Let's say that our test gives a sample average of 2017 hours for our bulbs and 1007 for the competitor's bulbs. The standard deviations are 3.2 hours for our bulbs and 3.5 hours for the competitor's bulbs.
Stated both algebraically and verbally, what would be the null and alternate hypotheses that we want to test? Why?
2nd part

(That last sentence is a bit of misdirection. By stating the question in that way, I need to consider the outcome of the experiment before stating the hypotheses to be tested. But consideration of the outcome is part of Step 5 of the 5-step hypothesis testing procedure)
Strictly speaking, you should always do Step 1 before Step 2, Step 2 before Step 3, Step 3 before Step 4, and Step 4 before Step 5. You should not do Step 5 before Step 1!
Or should you?
Suppose that, before you do the hypothesis test, you already knew the outcome of the light bulb experiment. Suppose that, instead of 1007 hours, the competitor's bulbs had lasted an average of 1009 hours. Can that tiny shift in the outcome have any effect on the design of our hypotheses? Should it?
Hint: The numbers 1007 and 1009 are radically different in impact compared to 2017 for our bulbs.

#### Solution Summary

This solution with step-by-step explanations provides a null and alternative hypothesis and provides the equation for the test statistic and how a tiny shift can impact it.

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