Word Flow: A keyboard alternative designed to work with one-handed, “thumb-swipe” typing while on the go. When given to students in middle school, the typing speed using Word Flow (measured in words per minute, wpm) averaged 20wpm with a standard deviation of 2.5 wpm and followed a normal distribution.

a. What percent of middle schoolers can type between 24.0 and 25.3 wpm? 

b. What proportion of middle schoolers can type either slower than 19 wpm or faster than 24 wpm

### Given Information:
- Mean typing speed (\(\mu\)) = 20 wpm
- Standard deviation (\(\sigma\)) = 2.5 wpm
- The distribution is normal.

### Part (a): What percent of middle schoolers can type between 24.0 and 25.3 wpm?

1. We need to find the **z-scores** for 24.0 wpm and 25.3 wpm using the formula:
   \[
   z = \frac{x - \mu}{\sigma}
   \]
   where \(x\) is the data point, \(\mu\) is the mean, and \(\sigma\) is the standard deviation.

   - For \(x = 24.0\):
     \[
     z_1 = \frac{24.0 - 20}{2.5} = \frac{4.0}{2.5} = 1.6
     \]
   - For \(x = 25.3\):
     \[
     z_2 = \frac{25.3 - 20}{2.5} = \frac{5.3}{2.5} = 2.12
     \]

2. Using a **z-table** or a calculator to find the proportion corresponding to these z-scores:
   - The cumulative probability for \(z = 1.6\) is approximately 0.9452.
   - The cumulative probability for \(z = 2.12\) is approximately 0.9830.

3. The percent of middle schoolers typing between 24.0 and 25.3 wpm is the difference between these two probabilities:
   \[
   P(24.0 < X < 25.3) = P(z_2) - P(z_1) = 0.9830 - 0.9452 = 0.0378
   \]
   Therefore, approximately **3.78%** of middle schoolers type between 24.0 and 25.3 wpm.

---

### Part (b): What proportion of middle schoolers can type either slower than 19 wpm or faster than 24 wpm?

1. Find the **z-scores** for 19 wpm and 24 wpm:
   - For \(x = 19.0\):
     \[
     z_3 = \frac{19.0 - 20}{2.5} = \frac{-1.0}{2.5} = -0.4
     \]
   - For \(x = 24.0\), we already found:
     \[
     z_1 = 1.6
     \]

2. Using the **z-table**:
   - The cumulative probability for \(z = -0.4\) is approximately 0.3446.
   - The cumulative probability for \(z = 1.6\) is 0.9452 (from Part a).

3. The proportion slower than 19 wpm:
   \[
   P(X < 19.0) = P(z_3) = 0.3446
   \]

4. The proportion faster than 24 wpm:
   \[
   P(X > 24.0) = 1 - P(z_1) = 1 - 0.9452 = 0.0548
   \]

5. The total proportion for typing either slower than 19 wpm or faster than 24 wpm is:
   \[
   P(X < 19.0 \text{ or } X > 24.0) = 0.3446 + 0.0548 = 0.3994
   \]
   Therefore, approximately **39.94%** of middle schoolers type either slower than 19 wpm or faster than 24 wpm.

---

### Summary:
- **(a)** Approximately **3.78%** of middle schoolers type between 24.0 and 25.3 wpm.
- **(b)** Approximately **39.94%** of middle schoolers type either slower than 19 wpm or faster than 24 wpm.

Would you like to explore this topic in more detail, or do you have any other questions?

### Follow-up Questions:
1. How would these calculations change if the mean typing speed were higher?
2. What if we wanted to calculate the percentage of students typing between 15 and 20 wpm?
3. Can you explain how z-scores help in comparing different data points?
4. How would the proportion change if the standard deviation increased?
5. What is the interpretation of the normal distribution curve in these problems?

### Tip:
The z-score is a powerful tool for comparing individual scores or data points to the overall distribution, allowing you to see how far they are from the mean in terms of standard deviations.

Word Flow: A keyboard alternative designed to work with one-handed, “thumb-swipe” typing while on the go. When given to students in middle school, the typing speed using Word Flow (measured in words per minute, wpm) averaged 20wpm with a standard deviation of 2.5 wpm and followed a normal distribution.

a. What percent of middle schoolers can type between 24.0 and 25.3 wpm?

b. What proportion of middle schoolers can type either slower than 19 wpm or faster than 24 wpm?

Explore the normal distribution of typing speeds among middle school students using the Word Flow keyboard. Learn how to calculate the percentage of students typing between 24.0 and 25.3 wpm, and find the proportion typing either slower than 19 wpm or faster than 24 wpm. Includes key concepts in statistics like z-scores and cumulative probability.

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