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Answer :
a. Greater than 77: [tex]\( 16\% \)[/tex]
b. Less than 56: [tex]\( 2.5\% \)[/tex]
c. Less than 77: [tex]\( 84\% \)[/tex]
d. Less than 84: [tex]\( 97.5\% \)[/tex]
e. Greater than 70: [tex]\( 50\% \)[/tex]
f. Less than 63: [tex]\( 15.87\% \)[/tex]
g. Greater than 56: [tex]\( 97.5\% \)[/tex] (complement of less than 56)
h. Between 63 and 84: [tex]\( 81.63\% \)[/tex]
To solve these problems using the 68-95-99.7 rule for a normal distribution with mean [tex]\(\mu = 70\)[/tex] and standard deviation [tex]\(\sigma = 7\)[/tex] we will calculate the percentages step by step.
Step-by-step calculations:
a. Greater than 77:
Calculate the z-score:
[tex]\[ z = \frac{77 - 70}{7} = 1 \][/tex]
The percentage of heart rates greater than 77 is the area to the right of [tex]\( z = 1 \)[/tex] on the standard normal distribution.
[tex]\[ \text{Percentage} = 100\% - \text{percentage within } z = 1 \text{ (which is } 84\%\text{)} \][/tex]
[tex]\[ \text{Percentage} = 100\% - 84\% = 16\% \][/tex]
b. Less than 56:
Calculate the z-score:
[tex]\[ z = \frac{56 - 70}{7} = -2 \][/tex]
The percentage of heart rates less than 56 is the area to the left of [tex]\( z = -2 \)[/tex] on the standard normal distribution.
[tex]\[ \text{Percentage} = \text{percentage within } z = -2 \text{ (which is } 2.5\%\text{)} \][/tex]
c. Less than 77:
Already calculated in part (a), but for completeness:
[tex]\[ \text{Percentage} = \text{percentage within } z = 1 \text{ (which is } 84\%\text{)} \][/tex]
d. Less than 84:
Calculate the z-score:
[tex]\[ z = \frac{84 - 70}{7} = 2 \][/tex]
The percentage of heart rates less than 84 is the area to the left of[tex]\( z = 2 \)[/tex] on the standard normal distribution.
[tex]\[ \text{Percentage} = \text{percentage within } z = 2 \text{ (which is } 97.5\%\text{)} \][/tex]
e. Greater than 70:
This is [tex]\( 50\% \)[/tex] because the mean divides the normal distribution symmetrically.
f. Less than 63:
Calculate the z-score:
[tex]\[ z = \frac{63 - 70}{7} = -1 \][/tex]
The percentage of heart rates less than 63 is the area to the left of [tex]\( z = -1 \)[/tex] on the standard normal distribution.
[tex]\[ \text{Percentage} = \text{percentage within } z = -1 \text{ (which is } 15.87\%\text{)} \][/tex]
g. Greater than 56:
Already calculated in part (b), but for completeness:
[tex]\[ \text{Percentage} = 100\% - \text{percentage within } z = -2 \text{ (which is } 2.5\%\text{)} \][/tex]
h. Between 63 and 84:
To find the percentage of heart rates between 63 and 84, we calculate the sum of percentages from parts (d) and (f):
[tex]\[ \text{Percentage} = \text{percentage within } z = 2 \text{ (less than 84)} - \text{percentage within } z = -1 \text{ (less than 63)} \[ \text{Percentage} = 97.5\% - 15.87\% \][/tex]
[tex]\[ \text{Percentage} = 81.63\% \][/tex]
These percentages reflect the distribution of heart rates based on the given mean and standard deviation, using the properties of the normal distribution.
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