s) The lifetimes of a certain type of light bulbs (in years) have pt distributio

Question

s) The lifetimes of a certain type of light bulbs (in years) have pt distribution. The d ensity function is given as 0, otherwise What is the probablity that a light bulb would fail by the end of the Sh year? a. b. What is the mean lifetime of this type of light bulbs? c. What is the median lifetime of this type of light bulbs? 13. (18pts) The weights of packets of cookies produced by a certain manufacturer have a Normal distribution with a mean of 202 g and a standard deviation of 3g. a. What is the probability that a packet of cookies weigh more than 205g? Find the weight value c such that 15% of packets weigh less than c. (This is the 0.15 quantile) If the stamped weight on the packet is 200g, do we need to reset the filling amount so that only 1% of the packets are underweight (less than the stamped weight)? If the answer is yes, what is the minimum mean filling amount (assuming the standard deviation is still 3g)? c.

Explanation / Answer

13. We have been given params of the normal distribution :

Mean = 202
Stdev = 3

a. P(X>205) = P(Z> 205-202/3) = P(Z>1) = .1587

b. P(x<c) = .15, so, c-202 / 3 = -1.035, So, c = -1.035*3+202 = 198.895

c. New mean = 200g, stdev = 3 gm. Only 1% are underweight if
Z = 200-205/3 = -5/3, P(Z=-5/3) = P(Z=-1.67) = .0475. 4.75% are underweight. Yes, you need to reset the filling amount so that only 1% of the packets are underweight.
To make only 1% underweight we have a Z of -2.33 ,which is X-205/3 = -2.33
X = -2.33*3+205 = 198g, should be the stamped weight on the pocket so that only 1% is underweight

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