What is the probability that height of students lies between 150cm and 170cm. Average height is 160cm with standard deviation of 10

To find the probability that the height of students lies between 150 cm and 170 cm, we'll use the properties of the normal distribution. The mean (μ) and standard deviation (σ) are given as follows:

- Mean (μ) = 160 cm
- Standard Deviation (σ) = 10 cm

### Step 1: Calculate the Z-scores
We first need to calculate the Z-scores for the heights 150 cm and 170 cm.

**Z-score for 150 cm:**

\[
Z = \frac{X - \mu}{\sigma} = \frac{150 - 160}{10} = \frac{-10}{10} = -1
\]

**Z-score for 170 cm:**

\[
Z = \frac{X - \mu}{\sigma} = \frac{170 - 160}{10} = \frac{10}{10} = 1
\]

### Step 2: Look up the Z-scores in the standard normal distribution table
Now, we find the probabilities corresponding to these Z-scores:

- The probability corresponding to \( Z = -1 \) is approximately 0.1587.
- The probability corresponding to \( Z = 1 \) is approximately 0.8413.

### Step 3: Calculate the probability between 150 cm and 170 cm
The probability that the height lies between 150 cm and 170 cm is the difference between these two probabilities:

\[
P(150 \leq X \leq 170) = 0.8413 - 0.1587 = 0.6826
\]

### Final Answer:
The probability that the height of students lies between 150 cm and 170 cm is **0.6826**, or **68.26%**.

Would you like to explore this concept further or have any other questions?

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Here are five related questions for further exploration:
1. What is the probability that a student's height is less than 150 cm?
2. What is the probability that a student's height is more than 170 cm?
3. What percentage of students are taller than 180 cm?
4. If the average height were 165 cm, how would the probability between 150 cm and 170 cm change?
5. What height corresponds to the 95th percentile in this distribution?

**Tip:** When dealing with normal distributions, it's useful to remember that approximately 68% of the data lies within one standard deviation of the mean, 95% within two, and 99.7% within three (the empirical rule).

Explore how to calculate the probability that student heights lie between 150 cm and 170 cm using normal distribution properties. Given an average height of 160 cm and a standard deviation of 10 cm, learn step-by-step calculations with Z-scores and probabilities.

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