Probability Question

2014-11-21 6:09 am

(a) A manufacturer produces LED light bulbs of mean lifetime 12 months. Assume that the lifetime of the light bulbs follows the normal distribution N(12, 2^2) with a standard deviation of 2 months. A light bulb is said to be acceptable if its lifetime is more than 9 months. A batch of 10 light bulbs was produced independently. Find the probability that there are 9 or more acceptable light bulbs in the batch.

(b) Later the manufacturer suspects that the mean lifetime µ of the light bulbs is no longer 12 months. He then conducted a survey by collecting 50 random samples of the light bulbs. The average lifetime of the 50 samples is found to be 11.4 months. Assuming that the lifetime standard deviation remains the same, construct a 95% confidence interval for the mean µ.

Normal Distribution Table:http://images.books24x7.com/bookimages/id_15231/p2000af46g2710001vpp_thm.jpg

回答 (1)

Yin

2014-11-21 7:10 am

✔ 最佳答案

(a)
X is Random Variable of lifetime
Y is Random Variable of the number of acceptable light bulbs
Z follows the Standard Normal Distribution .i.e. Z~N(0,1)

X ~ N(12,2^2)

P( X > 9)
= 1 - P( X <= 9)
= 1 - P( Z <= (9-12)/2)
= 1 - P( Z <= -1.5)
= 1 - 0.0668 (check the table)
= 0.9332

Y~Bin( 10, P( X > 9) )
Y~Bin( 10, 0.9332 )

P( Y >= 9)
=P( Y = 9) + P( Y=10)
唔記得左條公式，自己CHECK返

(b)
n = 50
bar(x)= 11.4
sd = 2
alpha = 1 - 95% = 0.05
z(0.025) = 1.96

By formula,
Confidence interval =
[ bar(x) - z(0.025) * sd / n^0.5 , bar(x) + z(0.025) * sd / n^0.5 ]
[ 11.4 - 1.96 * 2/ 50^0.5 ,11.4 + 1.96 * 2/ 50^0.5 ]
=[10.85,11.95]

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