Sure, let me provide a detailed solution and explain some statistics concepts through this example. It appears that the original question involves finding the probability associated with certain bounds in a normal distribution with given parameters. Let's break this down step-by-step.
### Given Data
- Sample Size ([tex]\(n\)[/tex]): 85
- Population Mean ([tex]\(\mu\)[/tex]): 22
- Population Standard Deviation ([tex]\(\sigma\)[/tex]): 13
- Lower Bound: 19
- Upper Bound: 23
We need to calculate the probability that the sample mean lies between 19 and 23.
### Step-by-Step Solution
1. Calculate the Standard Error of the Mean (SEM):
The standard error of the mean (SEM) is calculated using the formula:
[tex]\[
\text{SEM} = \frac{\sigma}{\sqrt{n}}
\][/tex]
With [tex]\(\sigma = 13\)[/tex] and [tex]\(n = 85\)[/tex]:
[tex]\[
\text{SEM} = \frac{13}{\sqrt{85}}
\][/tex]
2. Calculate the Z-Scores for the Lower and Upper Bounds:
Z-scores are calculated to standardize the bounds. The z-score formula is:
[tex]\[
z = \frac{X - \mu}{\text{SEM}}
\][/tex]
For the Lower Bound (19):
[tex]\[
z_{\text{lower}} = \frac{19 - 22}{\frac{13}{\sqrt{85}}}
\][/tex]
Evaluating, we get:
[tex]\[
z_{\text{lower}} = -2.1275871824522046
\][/tex]
For the Upper Bound (23):
[tex]\[
z_{\text{upper}} = \frac{23 - 22}{\frac{13}{\sqrt{85}}}
\][/tex]
Evaluating, we get:
[tex]\[
z_{\text{upper}} = 0.7091957274840682
\][/tex]
3. Calculate the Probability:
Using the standard normal distribution, we find the probabilities corresponding to these z-scores using the cumulative distribution function (CDF).
Let [tex]\( \Phi(z) \)[/tex] represent the CDF of the standard normal distribution.
The probability that the sample mean lies between the lower and upper bounds is given by:
[tex]\[
\Phi(z_{\text{upper}}) - \Phi(z_{\text{lower}})
\][/tex]
From the z-scores:
[tex]\[
\Phi(0.7091957274840682) - \Phi(-2.1275871824522046) = 0.7442128248197002
\][/tex]
Therefore, the probability that the sample mean is between 19 and 23 is approximately [tex]\(0.7442\)[/tex].
### Summary of Results
- Z-Score for Lower Bound (19): [tex]\(-2.1276\)[/tex]
- Z-Score for Upper Bound (23): [tex]\(0.7092\)[/tex]
- Probability that the Sample Mean is Between 19 and 23: [tex]\(0.7442\)[/tex]
These calculations tell us that there is about a 74.42% chance that the sample mean will fall between 19 and 23 when taking a random sample of size 85 from a population with a mean of 22 and a standard deviation of 13.
