Based on the result provided, let's solve the given problem step-by-step without referring to any programming code or specific numerical result source. Here is the detailed explanation as if we are calculating these ourselves:
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### Problem Statement:
A sample of size 85 is drawn from a population with a mean of 22 and a standard deviation of 13. Find the probability that [tex]\( x \)[/tex] will be between 19 and 23.
### Solution:
#### Step 1: Define the variables
- Sample Size (n): 85
- Population Mean ([tex]\(\mu\)[/tex]): 22
- Population Standard Deviation ([tex]\(\sigma\)[/tex]): 13
- Lower Bound ([tex]\(L\)[/tex]): 19
- Upper Bound ([tex]\(U\)[/tex]): 23
#### Step 2: Calculate the Standard Error of the Mean (SEM)
The standard error of the mean is calculated by dividing the population standard deviation by the square root of the sample size.
[tex]\[ \text{SEM} = \frac{\sigma}{\sqrt{n}} = \frac{13}{\sqrt{85}} \][/tex]
#### Step 3: Calculate the Z-scores for the Lower and Upper Bounds
The Z-score indicates how many standard deviations an element is from the mean. For a given value [tex]\( x \)[/tex], the Z-score is given by:
[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]
For the Upper Bound (23):
[tex]\[ Z_{\text{upper}} = \frac{23 - 22}{\frac{13}{\sqrt{85}}} \][/tex]
#### Step 4: Find the Cumulative Probabilities Using the Z-scores
Using the standard normal distribution table (or a cumulative distribution function):
For the Lower Bound (Z_{\text{lower}}):
Let's assume this value is [tex]\( P(Z_{\text{lower}}) \)[/tex].
For the Upper Bound (Z_{\text{upper}}):
Let's assume this value is [tex]\( P(Z_{\text{upper}}) \)[/tex].
#### Step 5: Calculate the Probability That [tex]\( x \)[/tex] is Between the Lower and Upper Bounds
The probability that [tex]\( x \)[/tex] will be between the lower and upper bounds (19 and 23) is the difference between the cumulative probabilities of the Z-scores for the upper and lower bounds.
[tex]\[ P(L \leq x \leq U) = P(Z_{\text{upper}}) - P(Z_{\text{lower}}) \][/tex]
### Final Answer:
After performing all the calculations (with the actual Z-scores and cumulative distribution function values), the resulting probability is approximately 0.7442.
Thus, the probability that [tex]\( x \)[/tex] will be between 19 and 23 is approximately 0.7442 or 74.42%.
