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

0.121

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 * (7)/(12)`

`\mu_x =1 * (7)/(2)`

`\mu_x =3.5`

The standard deviation is :

`\sigma _x = \sqrt{((12-6)/(12-1)) * 6 * ((7)/(12))* (1-(7)/(12))}`

`\sigma _x = \sqrt{((6)/(11)) * 6 * ((7)/(12))* ((5)/(12))}`

`\sigma _x = \sqrt{(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`