Sure, let me provide a detailed step-by-step solution to the original problem:
### Given Data:
We are provided with the following information about a sample:
- Sample size (n): 85
- Population mean (μ): 22
- Population standard deviation (σ): 13
- Lower bound of the interval: 19
- Upper bound of the interval: 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 population standard deviation and the sample size:
[tex]\[ \text{SEM} = \frac{\sigma}{\sqrt{n}} \][/tex]
Given:
[tex]\[ \sigma = 13 \][/tex]
[tex]\[ n = 85 \][/tex]
Substituting the values:
[tex]\[ \text{SEM} = \frac{13}{\sqrt{85}} \][/tex]
[tex]\[ \text{SEM} \approx 1.409 \][/tex] (approximately)
#### 2. Calculate the z-scores for the lower and upper bounds:
The z-score for a value [tex]\( x \)[/tex] in a population is calculated using the formula:
[tex]\[ z = \frac{x - \mu}{\text{SEM}} \][/tex]
For the lower bound (x = 19):
[tex]\[ z_{\text{lower}} = \frac{19 - 22}{1.409} \][/tex]
[tex]\[ z_{\text{lower}} \approx -2.128 \][/tex]
For the upper bound (x = 23):
[tex]\[ z_{\text{upper}} = \frac{23 - 22}{1.409} \][/tex]
[tex]\[ z_{\text{upper}} \approx 0.709 \][/tex]
#### 3. Calculate the probability:
To find the probability [tex]\( P \)[/tex] that a sample mean lies between the lower and upper bounds, we need the cumulative distribution function (CDF) values of the z-scores. The CDF value of a z-score represents the area under the standard normal curve to the left of that z-score.
Let [tex]\( \phi(z) \)[/tex] denote the CDF of the standard normal distribution.
The required probability:
[tex]\[ P(19 < \bar{x} < 23) = \phi(z_{\text{upper}}) - \phi(z_{\text{lower}}) \][/tex]
From standard normal distribution tables or using a CDF calculator:
[tex]\[ \phi(0.709) \approx 0.760 \][/tex]
[tex]\[ \phi(-2.128) \approx 0.016 \][/tex]
So, the probability [tex]\( P \)[/tex] is:
[tex]\[ P \approx 0.760 - 0.016 \][/tex]
[tex]\[ P \approx 0.744 \][/tex]
### Summary of Results:
- The z-score for the lower bound (19) is approximately -2.128.
- The z-score for the upper bound (23) is approximately 0.709.
- The probability that a sample mean falls between 19 and 23 is approximately 0.744 or 74.4%.
By following these steps, you can understand how the values were computed and gain insight into how such problems are solved using statistics.
