https://file.aiquickdraw.com/mathbot/file/17212619128242zwifr0x.png

To solve the second part of the problem, where we need to find the probability that the sample mean of 23 adults' systolic blood pressure will be between 120 and 122.7 mmHg, we will use the following steps:

1. **Identify the parameters:**
   - Mean (\(\mu\)): 120 mmHg
   - Standard deviation (\(\sigma\)): 5.6 mmHg
   - Sample size (\(n\)): 23

2. **Calculate the standard error of the mean (SEM):**
   \[
   \text{SEM} = \frac{\sigma}{\sqrt{n}} = \frac{5.6}{\sqrt{23}}
   \]

3. **Convert the given range to z-scores:**
   \[
   z = \frac{X - \mu}{\text{SEM}}
   \]
   For \(X = 120\):
   \[
   z_{120} = \frac{120 - 120}{\text{SEM}} = 0
   \]
   For \(X = 122.7\):
   \[
   z_{122.7} = \frac{122.7 - 120}{\text{SEM}}
   \]

4. **Find the probability corresponding to these z-scores using the standard normal distribution.**

Let's perform these calculations.

First, calculate the standard error of the mean:
\[
\text{SEM} = \frac{5.6}{\sqrt{23}} \approx 1.166
\]

Next, find the z-score for 122.7:
\[
z_{122.7} = \frac{122.7 - 120}{1.166} \approx 2.32
\]

Now, we find the probability corresponding to the z-scores:
\[
P(120 < \bar{X} < 122.7) = P(0 < Z < 2.32)
\]

Using the standard normal distribution table or a calculator, we find:
\[
P(Z < 2.32) \approx 0.9898
\]
\[
P(Z < 0) = 0.5
\]

Thus, the probability is:
\[
P(0 < Z < 2.32) = 0.9898 - 0.5 = 0.4898
\]

Therefore, the probability that the sample mean will be between 120 and 122.7 mmHg is approximately 0.4898.

If you have any questions or need further details, please let me know. Here are 5 related questions you might want to ask:

1. How do you find the z-score for a given value?
2. What is the standard normal distribution and how is it used?
3. How do you interpret the probability result in this context?
4. What assumptions must hold true for the normal distribution to apply?
5. Can you explain how the standard error of the mean is calculated?

**Tip:** When working with probabilities and the normal distribution, always double-check the z-scores and their corresponding probabilities to ensure accuracy.

Learn how to calculate the probability that the sample mean of 23 adults' systolic blood pressure will be between 120 and 122.7 mmHg. This problem involves statistics, normal distribution, and standard error of the mean. Suitable for undergraduate students studying statistics.

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