Question

Add a cell phone assembly plant, 80% of the cell phone keypad pass inspection. A random sample of 181 keypads is analyzed. Find the probability that less than 76 in the sample keypads pass inspection.

Answer 1

Answer:

0% probability that less than 76 in the sample keypads pass inspection.

Step-by-step explanation:

I am going to use the normal approximation to the binomial to solve this question.

Binomial probability distribution

Probability of exactly x sucesses on n repeated trials, with p probability.

Can be approximated to a normal distribution, using the expected value and the standard deviation.

The expected value of the binomial distribution is:

$E(X) = np$

The standard deviation of the binomial distribution is:

$\sqrt{V(X)} = \sqrt{np(1-p)}$

Normal probability distribution

Problems of normally distributed samples can be solved using the z-score formula.

In a set with mean $\mu$ and standard deviation $\sigma$, the zscore of a measure X is given by:

$Z = \frac{X - \mu}{\sigma}$

The Z-score measures how many standard deviations the measure is from the mean. After finding the Z-score, we look at the z-score table and find the p-value associated with this z-score. This p-value is the probability that the value of the measure is smaller than X, that is, the percentile of X. Subtracting 1 by the pvalue, we get the probability that the value of the measure is greater than X.

When we are approximating a binomial distribution to a normal one, we have that $\mu = E(X)$, $\sigma = \sqrt{V(X)}$.

In this problem, we have that:

$n = 181, p = 0.8$

So

$\mu = E(X) = np = 181*0.8 = 144.8$

$\sqrt{V(X)} = \sqrt{np(1-p)} = \sqrt{181*0.8*0.2} = 5.38$

Find the probability that less than 76 in the sample keypads pass inspection.

Using continuity correction, this is $P(X < 76 - 0.5) = P(X < 75.5)$, which is the pvalue of Z when X = 75.5.

So

$Z = \frac{X - \mu}{\sigma}$

$Z = \frac{75.5 - 144.8}{5.38}$

$Z = -12.88$

$Z = -12.88$ has a pvalue of 0

0% probability that less than 76 in the sample keypads pass inspection.