Sure, here is the detailed step-by-step solution:
### Given Problem
We are given:
- A sample size of 85
- Population mean ([tex]\(\mu\)[/tex]) = 22
- Population standard deviation ([tex]\(\sigma\)[/tex]) = 13
- Lower bound of the range = 19
- Upper bound of the range = 23
### To Find
We need to find the probability that a sample mean will lie between 19 and 23.
### Step-by-Step Solution
#### 1. Identify Parameters
We need to calculate the z-scores for the lower and upper bounds of the range. The z-score is given by:
[tex]\[ z = \frac{X - \mu}{\sigma / \sqrt{n}} \][/tex]
where:
- [tex]\( X \)[/tex] is the value for which we are calculating the z-score,
- [tex]\( \mu \)[/tex] is the population mean,
- [tex]\( \sigma \)[/tex] is the population standard deviation,
- [tex]\( n \)[/tex] is the sample size.
#### 2. Calculate the Z-scores
First, we calculate the z-score for the lower bound (19):
[tex]\[ z_{\text{lower}} = \frac{19 - 22}{13 / \sqrt{85}} \][/tex]
[tex]\[ \approx -2.1276 \][/tex]
Next, we calculate the z-score for the upper bound (23):
[tex]\[ z_{\text{upper}} = \frac{23 - 22}{13 / \sqrt{85}} \][/tex]
[tex]\[ \approx 0.7092 \][/tex]
#### 3. Use Z-scores to Find Probabilities
Using standard normal distribution tables or cumulative distribution function (often denoted as [tex]\(\Phi(z)\)[/tex]), we find the probabilities corresponding to these z-scores. The probability that the sample mean lies between these two z-scores is:
[tex]\[ P(19 \leq \overline{X} \leq 23) = \Phi(0.7092) - \Phi(-2.1276) \][/tex]
#### 4. Calculate the Probability
By looking up these z-scores in the standard normal distribution table or using computational tools, we find that:
[tex]\[ \Phi(0.7092) \approx 0.76 \][/tex]
[tex]\[ \Phi(-2.1276) \approx 0.0158 \][/tex]
Thus,
[tex]\[ P(19 \leq \overline{X} \leq 23) = 0.76 - 0.0158 \][/tex]
[tex]\[ \approx 0.7442 \][/tex]
### Result
Therefore, the probability that the sample mean lies between 19 and 23 is approximately 0.7442, or about 74.42%.
This step-by-step method shows the principles and calculations behind determining the probability using the standard normal distribution.
