Question

Question #786ba

Answer 1

`z^star = -0.67449`

Explanation:

Finding a z score with a certain percentage of the area under a normal distribution curve lying to the right of it is often referred to as finding a "right tail" distribution. This is because, visually, the area being referred to can be seen as the right-hand portion of the normal graph past a certain z-score of `z^star`, as shown here:

To do this, one will need to refer to either a z-score probability table, or use a computer algebra system (CAS) to determine the exact z-score needed.

We begin by letting `z^star` refer to the desired z-score that will achieve the stated goal. Thus, we are looking for this to be true:

`P(Z > z^star) = 0.75`

(With `Z` referring to all z-scores in the area of interest of the curve.)

Note that many/most of the available z-score tables provide the cumulative area probability, often known as the "left tail" distribution. This is not necessarily a problem, because for any given value `z^star` it is true that:

`P(Z < z^star) + P(Z > z^star) = 1`

`:. P(Z > z^star) = 1 - P(Z < z^star)`

For this problem, then, we need to find `z^star` such that:

`0.75 = 1 - P(Z < z^star)`

`P(Z < z^star) = 1 - 0.75`

`P(Z < z^star) = 0.25`

Thus we need a cumulative (aka "left-hand") area under the normal distribution curve of 0.25. To get this, you must use either a z-score table or an online/calculator tool.

One such table gives us two `z^star` values which are close to our target, but not quite exact:

`P(Z < -0.68) = 0.24825`
`P(Z < -0.67) = 0.25143`

The true value is between -0.68 and -0.67. A calculator provides a value of -0.67449.