Answer :
Based on the problem description and the provided numerical result, it seems there might be some confusion or mistakes in the text you've provided. Let's clarify each part of the problem step-by-step to better understand the context and the situation.
### Step 1: Understanding the Sample Size and Given Statistics
- 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
### Step 2: Calculating Z-Scores for the Lower and Upper Bounds
To determine the probability that the sample mean falls between the lower and upper bounds, we start by calculating the Z-scores for both bounds.
1. Z-score for the lower bound ([tex]\(X_L\)[/tex]):
- Formula for Z-score:
[tex]\[ Z = \frac{X_L - \mu}{\sigma / \sqrt{n}} \][/tex]
- Substituting the given values:
[tex]\[ Z_{lower} = \frac{19 - 22}{13 / \sqrt{85}} \approx -2.1276 \][/tex]
(Using the result provided) [tex]\( Z_{lower} \approx -2.1275871824522046 \)[/tex]
2. Z-score for the upper bound ([tex]\(X_U\)[/tex]):
- Formula for Z-score:
[tex]\[ Z = \frac{X_U - \mu}{\sigma / \sqrt{n}} \][/tex]
- Substituting the given values:
[tex]\[ Z_{upper} = \frac{23 - 22}{13 / \sqrt{85}} \approx 0.7092 \][/tex]
(Using the result provided) [tex]\( Z_{upper} \approx 0.7091957274840682 \)[/tex]
### Step 3: Using Z-Scores to Find the Probability
To find the probability that the sample mean lies between 19 and 23, we need the cumulative distribution function (CDF) values of the Z-scores:
[tex]\[ \text{Probability} = \text{CDF}(Z_{upper}) - \text{CDF}(Z_{lower}) \][/tex]
Where:
- [tex]\(\text{CDF}(Z_{lower})\)[/tex] is the area to the left of [tex]\( Z_{lower} \)[/tex]
- [tex]\(\text{CDF}(Z_{upper})\)[/tex] is the area to the left of [tex]\( Z_{upper} \)[/tex]
From the CDF for standard normal distribution:
- CDF value for [tex]\(Z_{lower} = -2.1275871824522046\)[/tex]
- CDF value for [tex]\(Z_{upper} = 0.7091957274840682\)[/tex]
Combining these results:
[tex]\[ \text{Probability} \approx \text{CDF}(0.7092) - \text{CDF}(-2.1276) \][/tex]
From the table, we find that:
[tex]\[ \text{CDF}(0.7092) \approx 0.76 \][/tex]
[tex]\[ \text{CDF}(-2.1276) \approx 0.016 \][/tex]
Thus, the final probability is:
[tex]\[ \text{Probability} \approx 0.76 - 0.016 = 0.7442 \][/tex]
### Conclusion
So, the probability that the sample mean ([tex]\(\bar{X}\)[/tex]) falls between 19 and 23 is approximately 0.7442 if we plug in the values straight from the statistical table's results. The final values for the Z-scores and probability are:
- [tex]\(Z_{lower} \approx -2.1275871824522046\)[/tex]
- [tex]\(Z_{upper} \approx 0.7091957274840682\)[/tex]
- [tex]\(\text{Probability} \approx 0.7442128248197002\)[/tex]
This concludes our solution to the problem.
### Step 1: Understanding the Sample Size and Given Statistics
- 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
### Step 2: Calculating Z-Scores for the Lower and Upper Bounds
To determine the probability that the sample mean falls between the lower and upper bounds, we start by calculating the Z-scores for both bounds.
1. Z-score for the lower bound ([tex]\(X_L\)[/tex]):
- Formula for Z-score:
[tex]\[ Z = \frac{X_L - \mu}{\sigma / \sqrt{n}} \][/tex]
- Substituting the given values:
[tex]\[ Z_{lower} = \frac{19 - 22}{13 / \sqrt{85}} \approx -2.1276 \][/tex]
(Using the result provided) [tex]\( Z_{lower} \approx -2.1275871824522046 \)[/tex]
2. Z-score for the upper bound ([tex]\(X_U\)[/tex]):
- Formula for Z-score:
[tex]\[ Z = \frac{X_U - \mu}{\sigma / \sqrt{n}} \][/tex]
- Substituting the given values:
[tex]\[ Z_{upper} = \frac{23 - 22}{13 / \sqrt{85}} \approx 0.7092 \][/tex]
(Using the result provided) [tex]\( Z_{upper} \approx 0.7091957274840682 \)[/tex]
### Step 3: Using Z-Scores to Find the Probability
To find the probability that the sample mean lies between 19 and 23, we need the cumulative distribution function (CDF) values of the Z-scores:
[tex]\[ \text{Probability} = \text{CDF}(Z_{upper}) - \text{CDF}(Z_{lower}) \][/tex]
Where:
- [tex]\(\text{CDF}(Z_{lower})\)[/tex] is the area to the left of [tex]\( Z_{lower} \)[/tex]
- [tex]\(\text{CDF}(Z_{upper})\)[/tex] is the area to the left of [tex]\( Z_{upper} \)[/tex]
From the CDF for standard normal distribution:
- CDF value for [tex]\(Z_{lower} = -2.1275871824522046\)[/tex]
- CDF value for [tex]\(Z_{upper} = 0.7091957274840682\)[/tex]
Combining these results:
[tex]\[ \text{Probability} \approx \text{CDF}(0.7092) - \text{CDF}(-2.1276) \][/tex]
From the table, we find that:
[tex]\[ \text{CDF}(0.7092) \approx 0.76 \][/tex]
[tex]\[ \text{CDF}(-2.1276) \approx 0.016 \][/tex]
Thus, the final probability is:
[tex]\[ \text{Probability} \approx 0.76 - 0.016 = 0.7442 \][/tex]
### Conclusion
So, the probability that the sample mean ([tex]\(\bar{X}\)[/tex]) falls between 19 and 23 is approximately 0.7442 if we plug in the values straight from the statistical table's results. The final values for the Z-scores and probability are:
- [tex]\(Z_{lower} \approx -2.1275871824522046\)[/tex]
- [tex]\(Z_{upper} \approx 0.7091957274840682\)[/tex]
- [tex]\(\text{Probability} \approx 0.7442128248197002\)[/tex]
This concludes our solution to the problem.
