Question

Z=1.23 z=0.86 WHAT is the area of the shaded region between the two

Answer 1

The area of the shaded region between the two z-scores can be found by subtracting the smaller area from the larger area between these two z-scores.

Step-by-step explanation:

The area of the shaded region between the two values z=1.23 and z=0.86 can be found by subtracting the smaller area from the larger area between these two z-scores.

Using the Z-table, we can find that the area to the left of z=1.23 is 0.8892 and the area to the left of z=0.86 is 0.8051. Therefore, the area between these two z-scores is 0.8892 - 0.8051 = 0.0841.

So, the area of the shaded region between z=1.23 and z=0.86 is 0.0841.

Answer 2

The area of the shaded region between
`\\ z = 1.23` and
`\\ z = 0.86` is
`\\ P(0.86 < z < 1.23) = 0.08554` or 8.554%.

Step-by-step explanation:

To solve this question, we need to find the corresponding probabilities for the standardized values (or z-scores) z = 1.23 and z = 0.86, and then subtract both to obtain the area of the shaded region between these two z-scores.

We need to having into account that a z-score is given by the following formula:

`\\ z = (x - \mu)/(\sigma)`

Where

• x is a raw score from the distribution that we want to standardize using [1].
• 
  `\\ \mu` is the mean of the normal distribution.
• 
  `\\ \sigma` is the standard deviation of the normal distribution.

A z-score indicates the distance of x from the mean in standard deviations units, where a positive value "tell us" that x is above
`\\ \mu`, and conversely, a negative that x is below
`\\ \mu`.

The standard normal distribution is a normal distribution with
`\\ \mu = 0` and
`\\ \sigma = 1`, and has probabilities for standardized values obtained using [1]. All these probabilities are tabulated in the standard normal table (available in any Statistical book or on the Internet).

Using the cumulative standard normal table, for
`\\ z = 1.23`, the corresponding cumulative probability is:

`\\ P(z<1.23) = 0.89065`

The steps are as follows:

1. Consult the cumulative standard table using z = 1.2 as an entry. Z-scores are in the first column of the mentioned table.
2. In the first row of it we have +0.00, +0.01, +0.02 and, finally, +0.03. The probability is the point that result from the intersection of z = 1.2 and +0.03 in the table, which is
   `\\ P(z<1.23) = 0.89065`.

Following the same procedure, the cumulative probability for
`\\ z = 0.86` is:

`\\ P(z<0.86) = 0.80511`

Subtracting both probabilities (because we need to know the area between these two values) we finally obtain the corresponding area between them (two z-scores):

`\\ P(0.86 < z < 1.23) = 0.89065 - 0.80511`

`\\ P(0.86 < z < 1.23) = 0.08554`

Therefore, the area of the shaded region between
`\\ z = 1.23` and
`\\ z = 0.86` is
`\\ P(0.86 < z < 1.23) = 0.08554` or 8.554%.

We can see this resulting area (red shaded area) in the graph below for a standard normal distribution,
`\\ N(0, 1)`, and
`\\ z = 0.86` and
`\\ z = 1.23`.