https://file.aiquickdraw.com/mathbot/file/1724590940209bk1kp176.jpg

To solve question 4 from the image you've provided, I'll need to read and interpret the content. Let me first extract and analyze the relevant information from the image.

Please hold on for a moment.It looks like the image is not very clear, and the text is quite small. However, I will try my best to read and interpret the content of question 4. 

From the visible text, question 4 appears to be related to a normal distribution problem involving mean (\(\mu\)) and standard deviation (\(\sigma\)).

**Question 4:**
Let's interpret the content step-by-step based on the visual information:

4. **If X is normally distributed with mean \(\mu = 8\) and standard deviation \(\sigma = 1.5\), find the probability that X is greater than 8.1.**

To solve this:

1. **Standardize the value:** Convert the value \(X = 8.1\) to a z-score using the formula:
   \[
   z = \frac{X - \mu}{\sigma}
   \]
   Here, \(X = 8.1\), \(\mu = 8\), and \(\sigma = 1.5\).

   \[
   z = \frac{8.1 - 8}{1.5} = \frac{0.1}{1.5} \approx 0.0667
   \]

2. **Find the probability:** We need to find \(P(X > 8.1)\), which corresponds to \(P(Z > 0.0667)\) for the standard normal distribution.

   To find this probability, use the standard normal table or a calculator:

   \[
   P(Z > 0.0667) = 1 - P(Z \leq 0.0667)
   \]

   Using a standard normal distribution table, \(P(Z \leq 0.0667) \approx 0.5266\).

   Therefore,

   \[
   P(Z > 0.0667) = 1 - 0.5266 = 0.4734
   \]

So, the probability that \(X > 8.1\) is approximately **0.4734**.

Would you like more details on this solution or have any other questions? 

Here are some follow-up questions that might help you understand this topic better:

1. What is the z-score and how is it used in probability calculations?
2. How do you interpret the standard normal distribution table?
3. Why do we subtract from 1 when finding \(P(Z > z)\) using the standard normal table?
4. Can you explain the significance of standard deviation in a normal distribution?
5. How do you calculate probabilities for non-standard normal distributions?

**Tip:** Remember, when dealing with probabilities in normal distributions, converting to z-scores simplifies the problem and allows you to use standard normal distribution tables effectively.

Learn how to calculate the probability for a normal distribution problem involving mean \(\mu = 8\) and standard deviation \(\sigma = 1.5\), where you need to find the probability that X is greater than 8.1. Detailed step-by-step solution with formulas and interpretation. Suitable for undergraduate-level students.

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