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Steps to finding the percentage of values in distribution between 170 and 172

To find the percentage of values in the distribution between 170 and 172, let's follow these steps using the **68-95-99.7 rule** for normal distribution.

### Given:
- **Mean (μ)** = 170
- **Standard Deviation (σ)** = 2

The **68-95-99.7 rule** tells us:
1. **68%** of the data falls within **±1σ** of the mean.
2. **95%** of the data falls within **±2σ** of the mean.
3. **99.7%** of the data falls within **±3σ** of the mean.

### Solution:
1. **Determine the Z-scores** for 170 and 172 in terms of standard deviations:
   - 170 is exactly the mean (0 standard deviations from the mean).
   - 172 is **1 standard deviation** above the mean (since \(172 = 170 + 2\)).

2. **Apply the 68% Rule**: 
   Since we know that 68% of the values lie within 1 standard deviation (from 168 to 172), half of this 68% will be between 170 and 172 because 170 is the center point.
   - \(\frac{68\%}{2} = 34\%\)

### Answer:
- **34%** of the values in the distribution fall between 170 and 172.

Would you like more detailed steps, or have any questions?

### Follow-up Questions:
1. How would you calculate the percentage of values between 168 and 170?
2. What percentage of values falls between 172 and 174?
3. Can we find the percentage of values between 170 and 174 without using a Z-table?
4. How does the 68-95-99.7 rule help in identifying the spread of data in normal distributions?
5. What percentage lies beyond 3 standard deviations from the mean?

### Tip:
The 68-95-99.7 rule is a quick way to estimate probabilities without calculating exact values using a Z-table.

A machine fills bags of candy. Due to slight irregularities in the operation of the machine, not every bag gets exactly the same number of pieces. Assume that the number of pieces per bag has a mean of 170 with a standard deviation of 2. Use the 68-95-99.7 rule to find the percentage of values in the distribution between 170 and 172.

Learn how to find the percentage of values between 170 and 172 in a normal distribution with a mean of 170 and a standard deviation of 2 using the 68-95-99.7 rule. This example covers normal distribution and probability, suitable for high school statistics students.

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