To solve the problem provided, it appears that we are dealing with a statistical problem involving a data sample, population mean, population standard deviation, and specific bounds within which we want to determine a probability. Here is the step-by-step explanation of how the solution is reached:
1. Understand the Given Data:
- Sample Size (n): 85
- Population Mean (μ): 22
- Population Standard Deviation (σ): 13
- Lower Bound: 19
- Upper Bound: 23
2. State the Problem:
We need to find the probability that a sample mean falls between 19 and 23 given the specified population parameters.
3. Determine the Z-scores for the Lower and Upper Bounds:
- The Z-score formula for a sample mean is: [tex]\( Z = \frac{(X - μ)}{\left(\frac{σ}{\sqrt{n}}\right)} \)[/tex]
- For the Lower Bound:
[tex]\[
Z_{lower} = \frac{(19 - 22)}{\left(\frac{13}{\sqrt{85}}\right)} = -2.1275871824522046
\][/tex]
- For the Upper Bound:
[tex]\[
Z_{upper} = \frac{(23 - 22)}{\left(\frac{13}{\sqrt{85}}\right)} = 0.7091957274840682
\][/tex]
4. Calculate the Cumulative Distribution Function (CDF) Values:
The CDF gives the probability that a standard normal variable is less than or equal to a specific value (the Z-score).
- CDF for [tex]\( Z_{upper} \)[/tex]:
[tex]\[
\text{CDF}(0.7091957274840682)
\][/tex]
- CDF for [tex]\( Z_{lower} \)[/tex]:
[tex]\[
\text{CDF}(-2.1275871824522046)
\][/tex]
5. Determine the Probability:
The probability that the sample mean is between the lower bound and the upper bound is the difference between the two CDF values.
- [tex]\[
\text{Probability} = \text{CDF}(0.7091957274840682) - \text{CDF}(-2.1275871824522046) = 0.7442128248197002
\][/tex]
Thus, the probability that the sample mean falls between 19 and 23 is approximately 0.7442, or 74.42%. This means there is a 74.42% chance that the sample mean from this population will lie within this specified range.
