Question 1197869: There is a 8% chance of a certain type of light bulb being defective and there are 500 of these bulbs randomly sampled. Use the normal approximation to the binomial to find the following probabilities rounded to 3 decimal places.
a. Find the probability that fewer than 50 of the bulbs are defective.
b. Find the probability that more than 50 of the bulbs are defective.
c. Find the probability that between 40 and 50, inclusive, of the bulbs are defective.
Found 2 solutions by ewatrrr, ikleyn:
Answer by ewatrrr(24785)   (Show Source):
You can put this solution on YOUR website! Binomial Distribution:
n = 500 , p = .08
Using the normal approximation and the NOted continuity correction factor.
(the continuity correction factor used as a Binomial Distribution is not continuous)
mean =500*.08= 40
sd =  = 6.07
Using TI or similarly an inexpensive calculator like an Casio fx-115 ES plus
P(x < 50 ) =P(x < 49.5 ), normpdf(-9999 49.5, 40, 6.07)= .941
P(x > 50) = P( x > 50.5) = normpdf(50.5,9999, 40, 6.07) = .042
P( 40 ≤ x ≤ 50) = = normcdf( 39.5,50.5, 40, 6.07) = .491
Answer by ikleyn(52780)  (Show Source):
You can put this solution on YOUR website! .
There is a 8% chance of a certain type of light bulb being defective
and there are 500 of these bulbs randomly sampled. Use the normal approximation
to the binomial to find the following probabilities rounded to 3 decimal places.
a. Find the probability that fewer than 50 of the bulbs are defective.
b. Find the probability that more than 50 of the bulbs are defective.
c. Find the probability that between 40 and 50, inclusive, of the bulbs are defective.
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You have a binomial distribution with large number of trials n = 500
and individual probability of success p = 0.08.
You want to approximate it by the normal distribution.
You should use the mean value m = p*n = 0.08*500 = 40 and standard deviation
SD =  =  = 6.0663.
Also, you should use the continuity correction factor.
About approximation of the binomial distribution by normal distribution and continuity correction factor
see your textbook and/or these Internet sources
https://www.statisticshowto.com/probability-and-statistics/binomial-theorem/normal-approximation-to-the-binomial/
https://online.stat.psu.edu/stat414/lesson/28/28.1
https://stats.libretexts.org/Courses/Las_Positas_College/Math_40%3A_Statistics_and_Probability/06%3A_Continuous_Random_Variables_and_the_Normal_Distribution/6.04%3A_Normal_Approximation_to_the_Binomial_Distribution
For calculations, you may use your calculator (function normcdf), or Excel spreadsheet (function NORMDIST);
or online calculator https://onlinestatbook.com/2/calculators/normal_dist.html
(a) In this case, you should find the area under the normal curve on the left from 49.5
(using the correction factor)
P(x < 50) = normcdf(-9999, 49.5, 40, 6.0663) = 0.941 (rounded).
(b) In this case, you should find the area under the normal curve on the right from 50.5
(using the correction factor)
P(x > 50) = normcdf(50.5, 9999, 40, 6.0663) = 0.0417 (rounded).
(c) In this case, you should calculate the area under the normal curve between 39.5 and 50.5
(using the correction factor)
P(40 <= x <= 50) = normcdf(39.5, 50.5, 40, 6.0663) = 0.491 (rounded).
Solved.
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If you are a beginner student in learning probability distributions, I advise you to start learning
this kind of computations using the online calculator, to which I referred above.
It provides a graphical support, so at each step you do understand what you are doing.
In addition, this graphical support prevents you from making mistakes.
When you learn enough this kind of computations, you can switch to your regular calculator,
but even then you may use the online calculator for checking purposes.
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