Stats Solutions

[Step-by-Step] A certain car model is produced with a 2.5% defect rate. That is, the probability that a car which comes off the assembly line has a defect

Question: A certain car model is produced with a 2.5% defect rate. That is, the
probability that a car which comes off the assembly line has a defect is 2.5%.
Use a normal distribution to estimate the probability that no more than 20,000
cars will be produced with defects when a million cars are manufactured.

(by a property of Binomial distributions) and also

\[\sigma (X)=\sqrt{np(1-p)}=\sqrt{1,000,000\times 0.025\times 0.975}=156.12\]

We use normal approximation to compute the probability that X is no more that 20,000. We have that

\[\Pr \left( X\le 20,000 \right)\approx \Pr \left( X\le 20,000.5 \right)\text{ (normal correction)}\] \[=\Pr \left( \frac{X-25,000}{156.12}\le \frac{20,000.5-25,000}{156.12} \right)=\Pr \left( Z\le -32.0234 \right)=0\]

where Z has a standard normal distribution. Recall that if the sample size is large enough, then

\[\frac{X-\mu \left( X \right)}{\sigma \left( X \right)}=\frac{X-25,000}{156.12}\]

has an approximate standard normal distribution.

(2)

Assume that the probability of the number of breakdowns of a cell phone of a
certain make in the first year of service is a Poisson distribution and the
standard deviation of that distribution is 2. What is the probability that a
phone of that make will break down between 2 and 4 times in the first year of
service ?

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Solution: The downloadable solution consists of 2 pages
Deliverable: Word Document


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