Question

Each of 12 refrigerators of a certain type has been returned to a distributor because of an audible, highpitched, oscillating noise when the refrigerators are running. Suppose that 7 of these refrigerators have a defective compressor and the other 5 have less serious problems. If the refrigerators are examined in random order, let X be the number among the first 6 examined that have a defective compressor. What is the probability that X exceeds its mean valueby more than 1 standard deviation?

Answer 1

Answer:

0.121

Step-by-step explanation:

From the given information:

Let X be a hypergeometric random variable. i.e X $\sim$ h (x;:6,7,12)

To determine the mean and the standard deviation of X of a hypergeometric random variable; we have:

$\mu_x = E(X)$

$\mu_x = 6 \times \dfrac{7}{12}$

$\mu_x =1 \times \dfrac{7}{2}$

$\mu_x =3.5$

The standard deviation is :

$\sigma _x = \sqrt{(\dfrac{12-6}{12-1}) \times 6 \times (\dfrac{7}{12})\times (1-\dfrac{7}{12})}$

$\sigma _x = \sqrt{(\dfrac{6}{11}) \times 6 \times (\dfrac{7}{12})\times (\dfrac{5}{12})}$

$\sigma _x = \sqrt{\dfrac{35}{44}}$

$\sigma _x = 0.892$

However, the inequality for the event showcasing how X exceeds its mean value by more than 1 standard deviation is :

$X \geq \mu_x +\sigma_x$

$X \geq 3.5 +0.892$

$X \geq 4.392$

$X \geq 5$

∴

$P(X \geq \mu_x + \sigma_x ) = P(X \geq 5)$

$P(X \geq \mu_x + \sigma_x ) = 0.121$