Did you ever buy an incandescent light bulb that failed (either burned out or did not work) the first time you turned the switch on? When you put a new bulb into a light fixture, you expect it to light, and most of the time it does. Consider 8-packs of 60-watt bulbs and let x be the number of bulbs in a pack that “fail” the first time they are used. If 0.02 of all bulbs of this type fail on their first use and each 8-pack is considered a random sample,

a. List the probability distribution and draw the histogram of x.

b. What is the probability that any one 8-pack has no bulbs that fail on first use?

c. What is the probability that any one 8-pack has no more than one bulb that fails on first use?

d. Find the mean and standard deviation of x.

e. What proportion of the distribution is between

   and   ?

f. What proportion of the distribution is between

  2 and   2?

g. How does this information relate to the empirical rule and Chebyshev’s theorem? Explain.

h. Use a computer to simulate testing 100 8-packs of bulbs and observing x, the number of failures per 8-pack. Describe how the information from the simulation compares with what was expected (answers to parts a–g describe the expected results).

i. Repeat part h several times. Describe how these results compare with those of parts a–g and with part h.