https://file.aiquickdraw.com/mathbot/file/1727690363880zaqecmrs.jpg

Let's solve the two problems in the image one by one:

### 1. **Problem 1: Compute \( P(Z < 1.67) \) using the standard normal curve.**

The standard normal distribution has a mean \( \mu = 0 \) and a standard deviation \( \sigma = 1 \). To find the probability \( P(Z < 1.67) \), we refer to the Z-table or use a cumulative distribution function (CDF) for the standard normal distribution.

Using the Z-table or a calculator:
\[
P(Z < 1.67) \approx 0.9525
\]
So, the probability \( P(Z < 1.67) \) is approximately **0.9525**, or **95.25%**.

---

### 2. **Problem 2: Cholesterol Distribution**

We are told that adult male cholesterol levels are normally distributed with:
- Mean \( \mu = 200 \)
- Standard deviation \( \sigma = 25 \)

We need to calculate the probabilities for cholesterol levels in two different ranges:

#### a. **For cholesterol less than 165 (11-12):**

We need to find \( P(X < 165) \), where \( X \) is the cholesterol level. To standardize this, we compute the Z-score:
\[
Z = \frac{165 - 200}{25} = \frac{-35}{25} = -1.4
\]
Using the Z-table, \( P(Z < -1.4) \approx 0.0808 \).

Thus, the probability that cholesterol is less than 165 is approximately **0.0808** or **8.08%**.

#### b. **For cholesterol between 165 and 220 (13-15):**

We need to find \( P(165 < X < 220) \). This involves calculating the Z-scores for both 165 and 220.

For 165:
\[
Z_1 = \frac{165 - 200}{25} = -1.4
\]
For 220:
\[
Z_2 = \frac{220 - 200}{25} = 0.8
\]
From the Z-table:
\[
P(Z < -1.4) \approx 0.0808 \quad \text{and} \quad P(Z < 0.8) \approx 0.7881
\]
Thus, the probability \( P(165 < X < 220) \) is:
\[
P(165 < X < 220) = P(Z < 0.8) - P(Z < -1.4) = 0.7881 - 0.0808 = 0.7073
\]
The probability that cholesterol is between 165 and 220 is approximately **0.7073** or **70.73%**.

---

### Graph Construction:
The graph should depict a normal distribution centered at 200, with points marked for cholesterol levels at 165 and 220, shading the areas under the curve corresponding to the computed probabilities.

Would you like to see more details about any specific part of this solution or have any further questions?

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### Related Questions:
1. How do you compute probabilities for standard normal distributions without a table?
2. What are the general steps to compute probabilities in a normal distribution?
3. How is the Z-score formula derived and why is it useful?
4. Can we compute probabilities for any range of values in a normal distribution?
5. How would the probabilities change if the standard deviation or mean were different?

### Tip:
Always visualize normal distribution problems with a graph, marking key Z-scores and areas under the curve for clarity.

Solve for the probability using the standard normal curve and for cholesterol levels distributed with a mean of 200 and standard deviation of 25.

Compute the probability using the standard normal curve (Z < 1.67) and find the probabilities of cholesterol levels less than 165 and between 165 and 220 for a normal distribution with mean 200 and standard deviation 25. Understand Z-scores and their application in probability calculations.

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