Alright, let's break down the problem step-by-step using the given results.
First, we are given the following information related to a statistical problem involving normal distribution:
- The sample size is 85.
- The population mean is 22.
- The population standard deviation is 13.
- The lower bound is 19.
- The upper bound is 23.
Using these parameters, our goal is to understand the related question and arrive at the final probability.
### Step-by-Step Solution:
1. Understanding the Distribution:
We know that we are working with a normal distribution characterized by a mean (μ) and standard deviation (σ). The central limit theorem tells us that for a sufficiently large sample size (n), the distribution of the sample mean will be approximately normal.
2. Compute the Standard Error (SE):
The standard error of the mean (SE) is derived from the population standard deviation (σ) and the sample size (n) using the formula:
[tex]\[ \text{SE} = \frac{\sigma}{\sqrt{n}} \][/tex]
Given:
[tex]\[ \sigma = 13 \][/tex]
[tex]\[ n = 85 \][/tex]
[tex]\[ \text{SE} = \frac{13}{\sqrt{85}} \][/tex]
3. Determine Z-Scores for the Bounds:
The z-score represents the number of standard errors a specific value (x) is away from the mean (μ). The z-score is calculated as:
[tex]\[ z = \frac{x - \mu}{\text{SE}} \][/tex]
For the lower bound (x = 19):
[tex]\[ z_{\text{lower}} = \frac{19 - 22}{\text{SE}} \][/tex]
For the upper bound (x = 23):
[tex]\[ z_{\text{upper}} = \frac{23 - 22}{\text{SE}} \][/tex]
Given the results:
[tex]\[ z_{\text{lower}} = -2.1275871824522046 \][/tex]
[tex]\[ z_{\text{upper}} = 0.7091957274840682 \][/tex]
4. Finding the Probability:
To find the probability that a sample mean falls between the lower and upper bounds, we need to find the area under the normal curve between these z-scores. This is done using the cumulative distribution function (CDF) of the standard normal distribution.
[tex]\[ \text{Probability} = \Phi(z_{\text{upper}}) - \Phi(z_{\text{lower}}) \][/tex]
Where [tex]\( \Phi(z) \)[/tex] is the CDF of the standard normal distribution.
Using the provided result:
[tex]\[ \text{Probability} = 0.7442128248197002 \][/tex]
### Conclusion:
So, the step-by-step solution involves calculating the z-scores for the given lower and upper bounds, and then finding the area between these z-scores using the CDF of the normal distribution, leading to the final probability:
[tex]\[ \text{Probability} = 0.7442128248197002 \][/tex]
This means there is approximately a 74.42% chance that the sample mean will fall between 19 and 23.
