Respuesta :
Answer:
The area of the shaded region between [tex] \\ z = 1.23[/tex] and [tex] \\ z = 0.86[/tex] is [tex] \\ P(0.86 < z < 1.23) = 0.08554[/tex] 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:
[tex] \\ z = \frac{x - \mu}{\sigma}[/tex]
Where
- x is a raw score from the distribution that we want to standardize using [1].
- [tex] \\ \mu[/tex] is the mean of the normal distribution.
- [tex] \\ \sigma[/tex] 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 [tex] \\ \mu[/tex], and conversely, a negative that x is below [tex] \\ \mu[/tex].
The standard normal distribution is a normal distribution with [tex] \\ \mu = 0[/tex] and [tex] \\ \sigma = 1[/tex], 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 [tex] \\ z = 1.23[/tex], the corresponding cumulative probability is:
[tex] \\ P(z<1.23) = 0.89065[/tex]
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 [tex] \\ P(z<1.23) = 0.89065[/tex].
Following the same procedure, the cumulative probability for [tex] \\ z = 0.86[/tex] is:
[tex] \\ P(z<0.86) = 0.80511[/tex]
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):
[tex] \\ P(0.86 < z < 1.23) = 0.89065 - 0.80511[/tex]
[tex] \\ P(0.86 < z < 1.23) = 0.08554[/tex]
Therefore, the area of the shaded region between [tex] \\ z = 1.23[/tex] and [tex] \\ z = 0.86[/tex] is [tex] \\ P(0.86 < z < 1.23) = 0.08554[/tex] or 8.554%.
We can see this resulting area (red shaded area) in the graph below for a standard normal distribution, [tex] \\ N(0, 1)[/tex], and [tex] \\ z = 0.86[/tex] and [tex] \\ z = 1.23[/tex].
