2.If we know that the sample mean for men’s height is 157cm, population standard deviation is 6cm, and the sample size is 15000 people,
a.How many men have height between 150cm and 164cm?

To determine how many men have a height between 150 cm and 164 cm, we can use the **normal distribution** because the population standard deviation is provided and the sample size is large. Here's the step-by-step process:

### Step 1: Define the problem

We are given:
- **Sample mean (μ)** = 157 cm
- **Population standard deviation (σ)** = 6 cm
- **Sample size (n)** = 15,000

We want to find how many men have heights between **150 cm** and **164 cm**.

### Step 2: Standardize the heights

We will convert the heights of 150 cm and 164 cm into z-scores. The formula for a z-score is:

\[
z = \frac{x - \mu}{\sigma}
\]

#### For 150 cm:

\[
z_{150} = \frac{150 - 157}{6} = \frac{-7}{6} = -1.17
\]

#### For 164 cm:

\[
z_{164} = \frac{164 - 157}{6} = \frac{7}{6} = 1.17
\]

### Step 3: Find the cumulative probability

Using standard normal distribution tables or a calculator, we can find the probability corresponding to these z-scores.

- For **\( z = -1.17 \)**, the cumulative probability is approximately **0.1210**.
- For **\( z = 1.17 \)**, the cumulative probability is approximately **0.8790**.

To find the probability of a man having a height between 150 cm and 164 cm, we subtract the cumulative probabilities:

\[
P(150 \leq X \leq 164) = P(z_{164}) - P(z_{150}) = 0.8790 - 0.1210 = 0.758
\]

### Step 4: Calculate the number of men

To find the number of men with heights between 150 cm and 164 cm, we multiply the probability by the total number of men:

\[
\text{Number of men} = 0.758 \times 15,000 = 11,370
\]

Thus, approximately **11,370 men** have heights between 150 cm and 164 cm.

---

Would you like more details on how the z-scores or probabilities were calculated? 

Here are 5 related questions:
1. How do we calculate probabilities for z-scores using a standard normal distribution table?
2. How do z-scores help in comparing values from different distributions?
3. What is the significance of the population standard deviation in this context?
4. How would the result change if the sample size was smaller, for example 100 people?
5. How can we use this method to find the probability of a man's height exceeding 170 cm?

**Tip:** When working with large datasets and normal distributions, z-scores make it easy to calculate probabilities for values within certain ranges.

If we know that the sample mean for men’s height is 157cm, population standard deviation is 6cm, and the sample size is 15000 people, how many men have height between 150cm and 164cm?

Learn how to calculate the number of men with heights between 150cm and 164cm using the normal distribution. This problem explores key statistical concepts, including z-scores, the standard normal distribution, and the empirical rule. Ideal for students in Grades 10-12.

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