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2 The pulse rates of adult men approach a normal
distribution with a mean of 80 bpm (beats per minute)
with standard deviation of 7 bpm .
a) What percent of the adults have pulse rate less
than 70 bpm ?
b) If 60 bpm to 100 bpm is known to be normal, how
many percent of adults have pulse rates that lies
on these area?

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UpStudy AI · Accepted Answer

Let's solve each part of the problem step by step using the properties of the normal distribution.

### Given:
- **Mean (μ):** 80 beats per minute (bpm)
- **Standard Deviation (σ):** 7 bpm

### Part a) What percent of the adults have a pulse rate less than 70 bpm?

We want to find the probability $$ P(X < 70) $$ where $$ X $$ is the pulse rate.

1. **Convert to the standard normal variable (Z-score):**

$$
   Z = \frac{X - \mu}{\sigma} = \frac{70 - 80}{7} = \frac{-10}{7} \approx -1.4286
   $$

2. **Find the corresponding probability for $$ Z = -1.4286 $$:**

Using standard normal distribution tables or a calculator:

$$
   P(Z < -1.4286) \approx 0.0764 \text{ or } 7.64\%
   $$

**Interpretation:** Approximately **7.6%** of adult men have a pulse rate less than 70 bpm.

### Part b) If 60 bpm to 100 bpm is known to be normal, what percent of adults have pulse rates that lie within this range?

We want to find the probability $$ P(60 < X < 100) $$.

1. **Convert the pulse rates to Z-scores:**

- For **60 bpm**:
     $$
     Z = \frac{60 - 80}{7} = \frac{-20}{7} \approx -2.8571
     $$
   
   - For **100 bpm**:
     $$
     Z = \frac{100 - 80}{7} = \frac{20}{7} \approx 2.8571
     $$

2. **Find the corresponding probabilities:**

- **For $$ Z = 2.8571 $$:**
     $$
     P(Z < 2.8571) \approx 0.9979 \text{ or } 99.79\%
     $$
   
   - **For $$ Z = -2.8571 $$:**
     $$
     P(Z < -2.8571) = 1 - P(Z < 2.8571) = 1 - 0.9979 = 0.0021 \text{ or } 0.21\%
     $$

3. **Calculate the probability between 60 bpm and 100 bpm:**

$$
   P(60 < X < 100) = P(Z < 2.8571) - P(Z < -2.8571) = 0.9979 - 0.0021 = 0.9958 \text{ or } 99.58\%
   $$

**Interpretation:** Approximately **99.6%** of adult men have pulse rates between 60 bpm and 100 bpm.

---

### **Summary:**
- **a)** About **7.6%** of adult men have a pulse rate less than 70 bpm.
- **b)** Approximately **99.6%** of adult men have pulse rates between 60 bpm and 100 bpm.
**Part a)** About **7.6%** of adult men have a pulse rate less than 70 bpm. **Part b)** Approximately **99.6%** of adult men have pulse rates between 60 bpm and 100 bpm.

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