An electric company has been selling a light bulb it advertises as having an average life of 1000 hours and standard deviation 250 hours. Assume that the lifetime of these light bulbs is normally distributed.

(a) (5 points)Suppose the company makes 40 cents on each light bulb they sell, out of which they must pay for the cost of possible replacement under warranty. Each time a light bulb is replaced under warranty, it costs the company 70 cents.The company wishes to make 33 cents per light bulb after possible replacement, how long should the warranty be?

****************************************************************************

[Use the following additional information for parts (b) and (c) in this question]

The company's research staff claims to have improved the bulb so that its average life is now significantly higher and the percentage of lightbulbs with lifetimes over 1000 hours is also significantly higher. To test this claim, a random sample of 400 new light bulbs is selected. This sample shows an average lifetime of 1030 hours with a standard deviation 200 hours, and 272 new light bulbs in the sample have lifetimes more than 1000 hours.

*****************************************************************************

(b) (4 points)Has the average lifetime of the new light bulbs gone up significantly as claimed? Justify your answer using a 95% confidence interval.

(c) (5 points)Has the percent of the new light bulbs with lifetime over 1000 hours gone up significantly as claimed? Justify your answer using a 95% confidence interval.