Question

Assume the resting heart rates for a sample of individuals are normally distributed with a mean of 75 and a standard deviation of 5. Use the 68-95-99.7 rule to find the following quantities.
a. The relative frequency of rates less than 85 using the 68-95-99.7 rule is _____.
b. The relative frequency of rates greater than 80 using the 68-95-99.7 rule is _____.
c. The relative frequency of rates between 65 and 75 using the 68-95-99.7 rule is _____.

Answer 1

Answer:

a) 0.975

b) 0.16

c) 0.475

Step-by-step explanation:

The Empirical Rule states that, for a normally distributed random variable:

68% of the measures are within 1 standard deviation of the mean.

95% of the measures are within 2 standard deviation of the mean.

99.7% of the measures are within 3 standard deviations of the mean.

In this problem, we have that:

Mean = 75

Standard deviation = 5

We also have that since the normal distribution is symmetric, 50% of the measures are below the mean and 50% are above.

a. The relative frequency of rates less than 85 using the 68-95-99.7 rule is

85 is two standard deviations above the mean.

So all the 50% below the mean is below 85

And of the 50% above the mean, 95% is less than 85. So

0.5 + 0.95*0.5 = 0.975

b. The relative frequency of rates greater than 80 using the 68-95-99.7 rule is

80 is one standard deviations above the mean.

Of the 50% above the mean, 68% is less than 80. So 100-68 = 32% is more than 80.

0.32*0.5 = 0.16.

c. The relative frequency of rates between 65 and 75 using the 68-95-99.7 rule is

Within the mean and two standard deviations below the mean.

So 95% of 50%

0.95*0.5 = 0.475