The resting heart rates for a sample of individuals are normally distributed with a mean of 

6868 

and a standard deviation of 

1111. 

Use the​ 68-95-99.7 rule to find the percentage of heart rates in each of the following categories. 

a. Greater than 7979

b. Less than 4646

c. Less than 7979

d. Less than 9090

e. Greater than 6868

f. Less than 5757

g. Greater than 4646

h. Between 

5757 

and 9090

Question content area bottom

Part 1

a.**** The percentage of heart rates greater than 

7979 

is 

enter your response here​%.

​(Round to one decimal place as​ needed.)

Part 2

b. The percentage of heart rates less than 

4646 

is 

enter your response here​%.

​(Round to one decimal place as​ needed.)

Part 3

c.**** The percentage of heart rates less than 

7979 

is 

enter your response here​%.

​(Round to one decimal place as​ needed.)

Part 4

d.**** The percentage of heart rates less than 

9090 

is 

enter your response here​%.

​(Round to one decimal place as​ needed.)

Part 5

e.**** The percentage of heart rates greater than 

6868 

is 

enter your response here​%.

​(Round to one decimal place as​ needed.)

Part 6

f.**** The percentage of heart rates less than 

5757 

is 

enter your response here​%.

​(Round to one decimal place as​ needed.)

Part 7

g. The percentage of heart rates greater than 

4646 

is 

enter your response here​%.

​(Round to one decimal place as​ needed.)

Part 8

h.**** The percentage of heart rates between 

5757 

and 

9090 

is 

enter your response here​%.

​(Round to one decimal place as​ needed.)

We can use the 68-95-99.7 rule (Empirical Rule) to solve this problem. This rule tells us that for a normally distributed dataset:
- 68% of the data falls within 1 standard deviation of the mean.
- 95% falls within 2 standard deviations.
- 99.7% falls within 3 standard deviations.

Given:
- Mean \( \mu = 68 \)
- Standard deviation \( \sigma = 11 \)

Let's calculate the values at 1, 2, and 3 standard deviations from the mean:

- \( \mu + \sigma = 68 + 11 = 79 \)
- \( \mu - \sigma = 68 - 11 = 57 \)
- \( \mu + 2\sigma = 68 + 2(11) = 90 \)
- \( \mu - 2\sigma = 68 - 2(11) = 46 \)
- \( \mu + 3\sigma = 68 + 3(11) = 101 \)
- \( \mu - 3\sigma = 68 - 3(11) = 35 \)

Now, we can use these values to find the percentages for each category.

### Part 1: a. Greater than 79
79 is 1 standard deviation above the mean. According to the 68-95-99.7 rule:
- 68% of the data falls within 1 standard deviation (between 57 and 79).
- That leaves 32% of the data outside this range, split equally on both sides of the mean.
- Half of that 32% (i.e., 16%) lies above 79.

Thus, the percentage of heart rates greater than 79 is **16%**.

### Part 2: b. Less than 46
46 is 2 standard deviations below the mean. According to the rule:
- 95% of the data falls within 2 standard deviations (between 46 and 90).
- That leaves 5% of the data outside this range, with half of it (i.e., 2.5%) lying below 46.

Thus, the percentage of heart rates less than 46 is **2.5%**.

### Part 3: c. Less than 79
Since 79 is 1 standard deviation above the mean:
- 68% of the data falls within 1 standard deviation.
- So, 34% lies between the mean and 79.
- Adding the 50% of the data below the mean gives us 50% + 34% = **84%** of heart rates less than 79.

### Part 4: d. Less than 90
90 is 2 standard deviations above the mean. According to the rule:
- 95% of the data lies within 2 standard deviations.
- So, 95% of the heart rates are less than 90.

Thus, the percentage of heart rates less than 90 is **97.5%**.

### Part 5: e. Greater than 68
Since 68 is the mean, 50% of the data lies above the mean.

Thus, the percentage of heart rates greater than 68 is **50%**.

### Part 6: f. Less than 57
57 is 1 standard deviation below the mean. Using the same logic as in part (a):
- 16% of the data lies below 57.

Thus, the percentage of heart rates less than 57 is **16%**.

### Part 7: g. Greater than 46
46 is 2 standard deviations below the mean. Since only 2.5% of the data lies below 46:
- 100% - 2.5% = **97.5%** of heart rates are greater than 46.

### Part 8: h. Between 57 and 90
57 is 1 standard deviation below the mean, and 90 is 2 standard deviations above the mean:
- From the rule, 68% of the data lies within 1 standard deviation (between 57 and 79).
- An additional 13.5% lies between 79 and 90 (within the second standard deviation).

Thus, the percentage of heart rates between 57 and 90 is \( 68\% + 13.5\% = 81.5\% \).

### Summary:
a. Greater than 79: **16%**  
b. Less than 46: **2.5%**  
c. Less than 79: **84%**  
d. Less than 90: **97.5%**  
e. Greater than 68: **50%**  
f. Less than 57: **16%**  
g. Greater than 46: **97.5%**  
h. Between 57 and 90: **81.5%**

Would you like any clarifications or more details on these calculations?

---

Here are 5 related questions to expand your understanding:
1. What is the probability of a heart rate falling between 46 and 57?
2. How would the percentages change if the standard deviation was 15 instead of 11?
3. What is the percentage of heart rates between 68 and 90?
4. What percentage of heart rates fall outside of 3 standard deviations?
5. How would you calculate percentages for values not at whole standard deviations?

**Tip:** Always visualize a normal distribution curve when applying the Empirical Rule. It helps to understand the distribution of data across standard deviations.

The resting heart rates for a sample of individuals are normally distributed with a mean of 68 and a standard deviation of 11. Use the 68-95-99.7 rule to find the percentage of heart rates in each of the following categories:
a. Greater than 79
b. Less than 46
c. Less than 79
d. Less than 90
e. Greater than 68
f. Less than 57
g. Greater than 46
h. Between 57 and 90

Learn how to apply the 68-95-99.7 rule to calculate the distribution of resting heart rates. This example uses a mean of 68 and a standard deviation of 11 to find the percentages for heart rates in categories like 'greater than 79' and 'less than 46'. Perfect for students in Grades 10-12.

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