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Sagot :
Sure! Let's analyze and determine the probabilities for each of the given categories based on a normal distribution with a mean of 66 and a standard deviation of 6.
### Step-by-step solutions:
#### a. Percentage of heart rates greater than 72
Firstly, let's find the Z-score for 72. The Z-score is calculated using the formula:
[tex]\[ Z = \frac{X - \mu}{\sigma} \][/tex]
where [tex]\( X \)[/tex] is the value, [tex]\( \mu \)[/tex] is the mean, and [tex]\( \sigma \)[/tex] is the standard deviation.
[tex]\[ Z = \frac{72 - 66}{6} = 1 \][/tex]
Using the cumulative probability from the Z-table, the area to the left of [tex]\( Z \)[/tex] = 1 is approximately 0.8413 or 84.13%. Therefore, the percentage of heart rates greater than 72 is:
[tex]\[ 100\% - 84.13\% = 15.87\% \][/tex]
#### c. Percentage of heart rates less than 72
Using the Z-score calculated above for 72 ([tex]\( Z = 1 \)[/tex]), the cumulative probability to the left of this Z-score is approximately 84.13%. Hence, the percentage of heart rates less than 72 is:
[tex]\[ 84.13\% \][/tex]
#### e. Percentage of heart rates greater than 66
The Z-score for 66 is:
[tex]\[ Z = \frac{66 - 66}{6} = 0 \][/tex]
The cumulative probability to the left of [tex]\( Z = 0 \)[/tex] is 50%. Hence, the percentage of heart rates greater than or equal to 66 is:
[tex]\[ 100\% - 50\% = 50\% \][/tex]
#### g. Percentage of heart rates greater than 54
Calculate the Z-score for 54:
[tex]\[ Z = \frac{54 - 66}{6} = -2 \][/tex]
The cumulative probability to the left of [tex]\( Z = -2 \)[/tex] is approximately 2.28%. Therefore, the percentage of heart rates greater than 54 is:
[tex]\[ 100\% - 2.28\% = 97.72\% \][/tex]
#### b. Percentage of heart rates less than 54
Using the Z-score calculated for 54 ([tex]\( Z = -2 \)[/tex]), the cumulative probability to the left is about 2.28%. Thus, the percentage of heart rates less than 54 is:
[tex]\[ 2.28\% \][/tex]
#### d. Percentage of heart rates less than 78
Calculate the Z-score for 78:
[tex]\[ Z = \frac{78 - 66}{6} = 2 \][/tex]
The cumulative probability to the left of [tex]\( Z = 2 \)[/tex] is about 97.72%. Hence, the percentage of heart rates less than 78 is:
[tex]\[ 97.72\% \][/tex]
#### f. Percentage of heart rates less than 60
Calculate the Z-score for 60:
[tex]\[ Z = \frac{60 - 66}{6} = -1 \][/tex]
The cumulative probability to the left of [tex]\( Z = -1 \)[/tex] is about 15.87%. Therefore, the percentage of heart rates less than 60 is:
[tex]\[ 15.87\% \][/tex]
#### h. Percentage of heart rates between 60 and 78
For this, we need the cumulative probabilities for both 60 and 78 calculated earlier:
- The probability of being less than 78 is approximately 97.72%.
- The probability of being less than 60 is approximately 15.87%.
Thus, the percentage of heart rates between 60 and 78 is:
[tex]\[ 97.72\% - 15.87\% = 81.85\% \][/tex]
### Summary:
Here are the percentages for each category:
- Greater than 72: 15.87%
- Less than 72: 84.13%
- Greater than 66: 50.00%
- Greater than 54: 97.72%
- Less than 54: 2.28%
- Less than 78: 97.72%
- Less than 60: 15.87%
- Between 60 and 78: 81.85%
### Step-by-step solutions:
#### a. Percentage of heart rates greater than 72
Firstly, let's find the Z-score for 72. The Z-score is calculated using the formula:
[tex]\[ Z = \frac{X - \mu}{\sigma} \][/tex]
where [tex]\( X \)[/tex] is the value, [tex]\( \mu \)[/tex] is the mean, and [tex]\( \sigma \)[/tex] is the standard deviation.
[tex]\[ Z = \frac{72 - 66}{6} = 1 \][/tex]
Using the cumulative probability from the Z-table, the area to the left of [tex]\( Z \)[/tex] = 1 is approximately 0.8413 or 84.13%. Therefore, the percentage of heart rates greater than 72 is:
[tex]\[ 100\% - 84.13\% = 15.87\% \][/tex]
#### c. Percentage of heart rates less than 72
Using the Z-score calculated above for 72 ([tex]\( Z = 1 \)[/tex]), the cumulative probability to the left of this Z-score is approximately 84.13%. Hence, the percentage of heart rates less than 72 is:
[tex]\[ 84.13\% \][/tex]
#### e. Percentage of heart rates greater than 66
The Z-score for 66 is:
[tex]\[ Z = \frac{66 - 66}{6} = 0 \][/tex]
The cumulative probability to the left of [tex]\( Z = 0 \)[/tex] is 50%. Hence, the percentage of heart rates greater than or equal to 66 is:
[tex]\[ 100\% - 50\% = 50\% \][/tex]
#### g. Percentage of heart rates greater than 54
Calculate the Z-score for 54:
[tex]\[ Z = \frac{54 - 66}{6} = -2 \][/tex]
The cumulative probability to the left of [tex]\( Z = -2 \)[/tex] is approximately 2.28%. Therefore, the percentage of heart rates greater than 54 is:
[tex]\[ 100\% - 2.28\% = 97.72\% \][/tex]
#### b. Percentage of heart rates less than 54
Using the Z-score calculated for 54 ([tex]\( Z = -2 \)[/tex]), the cumulative probability to the left is about 2.28%. Thus, the percentage of heart rates less than 54 is:
[tex]\[ 2.28\% \][/tex]
#### d. Percentage of heart rates less than 78
Calculate the Z-score for 78:
[tex]\[ Z = \frac{78 - 66}{6} = 2 \][/tex]
The cumulative probability to the left of [tex]\( Z = 2 \)[/tex] is about 97.72%. Hence, the percentage of heart rates less than 78 is:
[tex]\[ 97.72\% \][/tex]
#### f. Percentage of heart rates less than 60
Calculate the Z-score for 60:
[tex]\[ Z = \frac{60 - 66}{6} = -1 \][/tex]
The cumulative probability to the left of [tex]\( Z = -1 \)[/tex] is about 15.87%. Therefore, the percentage of heart rates less than 60 is:
[tex]\[ 15.87\% \][/tex]
#### h. Percentage of heart rates between 60 and 78
For this, we need the cumulative probabilities for both 60 and 78 calculated earlier:
- The probability of being less than 78 is approximately 97.72%.
- The probability of being less than 60 is approximately 15.87%.
Thus, the percentage of heart rates between 60 and 78 is:
[tex]\[ 97.72\% - 15.87\% = 81.85\% \][/tex]
### Summary:
Here are the percentages for each category:
- Greater than 72: 15.87%
- Less than 72: 84.13%
- Greater than 66: 50.00%
- Greater than 54: 97.72%
- Less than 54: 2.28%
- Less than 78: 97.72%
- Less than 60: 15.87%
- Between 60 and 78: 81.85%
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