document.write( "Question 825716: Just when I thought I was begninng to understand probability I came across this problem. I'm really struggling with what \"N\" is.
\n" ); document.write( "A manufacturer of valves admits that its quality control has gone radically “downhill” such that currently the probability of producing a defective valve is 0.50. If it manufactures 1 million valves in a month and you randomly sample from these valves 10,000 samples, each composed of 15 valves.\r
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Algebra.Com's Answer #497542 by math-vortex(648)\"\" \"About 
You can put this solution on YOUR website!
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document.write( "Hi, there--\r\n" );
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document.write( "THE PROBLEM:\r\n" );
document.write( "A manufacturer of valves admits that its quality control has gone radically “downhill” such \r\n" );
document.write( "that currently the probability of producing a defective valve is 0.50. If it manufactures \r\n" );
document.write( "1 million valves in a month and you randomly sample from these valves 10,000 samples, \r\n" );
document.write( "each composed of 15 valves.\r\n" );
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document.write( "a. In how many samples would you expect to find exactly 13 good valves?\r\n" );
document.write( "b. In how many samples would you expect to find at least 13 good valves?\r\n" );
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document.write( "A SOLUTION:\r\n" );
document.write( "I agree that it's sometimes difficult to figure out what variables and values to use. \r\n" );
document.write( "I would treat this like a Binomial Distribution with X ~ Binom (15, 0.50)\r\n" );
document.write( "n = sample size = 15\r\n" );
document.write( "p = probability that randomly selected valve is good (not defective) = 0.50\r\n" );
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document.write( "a. In how many samples would you expect to find exactly 13 good values?\r\n" );
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document.write( "P(X=13) = (15C13) * (0.50)^13* (0.50)^2 = 105 * 0.000122 * 0.25 = 0.0032043\r\n" );
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document.write( "So the probability of finding exactly 13 good valves in your 15-valve sample is\r\n" );
document.write( "0.0032043457. So, out of 10,000 samples of 15 valves, we'd expect to see 32 samples with exactly 13 good values.\r\n" );
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document.write( "b. In home many samples would you expect to find at least 13 good valves?\r\n" );
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document.write( "P(X≥13) = P(X=13) + P(X=14) + P(X=15)\r\n" );
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document.write( "By the same method:\r\n" );
document.write( "P(X=14) =0.0004578\r\n" );
document.write( "P(X=15) =0.0000305\r\n" );
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document.write( "P(X≥13) = 0.0032043 + 0.0004578 + 0.0000305 = 0.0036926\r\n" );
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document.write( "So, out of 10,000 such samples, we expect to see 37 samples with at least 13 good valves.\r\n" );
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document.write( "Hope this helps!\r\n" );
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document.write( "Mrs. Figgy\r\n" );
document.write( "math.in.the.vortex@gmail.com\r\n" );
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