Let's start by analyzing the question and breaking down each part into meaningful and accurate steps to derive the result.
Given data:
- 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
We need to find the probability that a sample mean falls between 19 and 23.
Step-by-step solution:
1. Find the Standard Error:
The standard error of the mean (SEM) is given by the formula:
[tex]\[
\text{SEM} = \frac{\sigma}{\sqrt{n}}
\][/tex]
Plugging in the given values:
[tex]\[
\text{SEM} = \frac{13}{\sqrt{85}}
\][/tex]
2. Calculate the Z-scores for the Lower and Upper Bounds:
The Z-score for a value [tex]\( X \)[/tex] in a normal distribution is calculated as:
[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]
3. Look up the Cumulative Distribution Function (CDF):
The normal distribution table (or a computational tool) provides the cumulative probability up to a given Z-score.
- Determine the cumulative probability for both Z-scores.
4. Probability Calculation:
The probability that the sample mean lies between 19 and 23 is the difference in the cumulative probabilities for the upper and lower bounds.
[tex]\[
P(19 < \bar{X} < 23) = \text{CDF}(Z_{\text{upper}}) - \text{CDF}(Z_{\text{lower}})
\][/tex]
5. Final Numerical Values:
By calculating the Z-scores explicitly:
[tex]\[
Z_{\text{lower}} \approx -2.1276
\][/tex]
[tex]\[
Z_{\text{upper}} \approx 0.7092
\][/tex]
- From standard normal CDF tables, or using computational tools:
- The CDF value at [tex]\( Z_{\text{lower}} \approx -2.1276 \)[/tex] is approximately 0.0167.
- The CDF value at [tex]\( Z_{\text{upper}} \approx 0.7092 \)[/tex] is approximately 0.7609.
Thus, the probability [tex]\( P \)[/tex] is given by:
[tex]\[
P(19 < \bar{X} < 23) = 0.7609 - 0.0167 = 0.7442
\][/tex]
So, the probability that the sample mean lies between 19 and 23 is approximately [tex]\(0.7442\)[/tex], or 74.42%.
This answers the question requiring the calculation within this specific range.
