Mathematics, 26.06.2020 16:01 ineedhelp2285

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

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Answer from: jakeagilbert

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:

$\\ z = \frac{x - \mu}{\sigma}$

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:

$\\ P(z$

The steps are as follows:

Consult the cumulative standard table using z = 1.2 as an entry. Z-scores are in the first column of the mentioned table. 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 $\\ P(z$ .

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

$\\ P(z$

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 $\\ z = 1.23$ .