Let's solve the problem step-by-step.
We are given:
- A normal distribution with a mean ([tex]\(\mu\)[/tex]) of 45.8 and a standard deviation ([tex]\(\sigma\)[/tex]) of 27.6.
- We need to find the probability that a randomly selected value [tex]\(X\)[/tex] is between -6.6 and 106.5.
The general approach involves the following steps:
1. Calculate the z-scores for the lower and upper bounds.
2. Find the cumulative probability values corresponding to these z-scores.
3. Compute the probability of [tex]\(X\)[/tex] falling between -6.6 and 106.5.
### Step 1: Calculate the z-scores
The z-score is calculated by the formula:
[tex]\[ z = \frac{X - \mu}{\sigma} \][/tex]
For the lower bound ([tex]\(X = -6.6\)[/tex]):
[tex]\[
z_{\text{lower}} = \frac{-6.6 - 45.8}{27.6} \approx -1.8986
\][/tex]
For the upper bound ([tex]\(X = 106.5\)[/tex]):
[tex]\[
z_{\text{upper}} = \frac{106.5 - 45.8}{27.6} \approx 2.1993
\][/tex]
### Step 2: Find the cumulative probability values
The cumulative distribution function (CDF) gives the probability that a standard normal variable is less than or equal to a given value [tex]\(z\)[/tex].
Using standard normal distribution tables or a statistical software for precision:
[tex]\[
P(Z \le z_{\text{lower}}) \approx P(Z \le -1.8986) \approx 0.0289
\][/tex]
[tex]\[
P(Z \le z_{\text{upper}}) \approx P(Z \le 2.1993) \approx 0.9862
\][/tex]
### Step 3: Compute the probability
The probability that [tex]\(X\)[/tex] is between -6.6 and 106.5 is:
[tex]\[
P(-6.6 < X < 106.5) = P(Z \le z_{\text{upper}}) - P(Z \le z_{\text{lower}})
\][/tex]
Plugging in the values:
[tex]\[
P(-6.6 < X < 106.5) = 0.9862 - 0.0289 = 0.9573
\][/tex]
Thus, the probability that a randomly selected value from this distribution falls between -6.6 and 106.5 is approximately:
[tex]\[
0.9573
\][/tex]
Therefore, the given probability [tex]\(P(-6.6 < X < 106.5)\)[/tex] is approximately [tex]\(\boxed{0.9573}\)[/tex].
