Answer:
At least 31.5 years old: Approximately 4116 people
Less than 31.5 years old: Approximately 784 people
Step-by-step explanation:
To use the 68-95-99.7 Rule (Empirical Rule) for a normally distributed dataset, first, we need to convert the ages into months because the standard deviation is given in months. The mean age is 34 years, which is equivalent to [tex]( 34 \times 12 = 408 )[/tex] months.
Now, we want to find how many people are at least 31.5 years old and how many are less than that. First, we convert 31.5 years into months:
[tex]( 31.5 \times 12 = 378 )[/tex] months.
Next, let's determine how many standard deviations away 378 months is from the mean (408 months):
[tex]( Z = \frac{X - \mu}{\sigma} )[/tex]
Where:
( Z ) is the Z-score,
( X ) is the value in the dataset (378 months),
[tex]( \mu )[/tex] is the mean (408 months),
[tex]( \sigma )[/tex] is the standard deviation (30 months).
[tex]( Z = \frac{378 - 408}{30} = \frac{-30}{30} = -1 )[/tex]
A Z-score of -1 falls within the range that covers 68% of the data (34% on each side of the mean) plus the 50% below the mean, totaling 84%. This means that 84% of the data falls at or above a Z-score of -1.
Now let's calculate the number of people that corresponds to:
[tex]( 0.84 \times 4900 \approx 4116 )[/tex] people.
This means that approximately 4116 people are at least 31.5 years old.
For part (b), since the curve is symmetrical, the number of people less than 31.5 years old would be the remaining percentage:
( 100% - 84% = 16% )
Let's calculate the number of people that corresponds to:
[tex]( 0.16 \times 4900 \approx 784 )[/tex] people
This means that approximately 784 people are less than 31.5 years old.
Summary:
At least 31.5 years old: Approximately 4116 people
Less than 31.5 years old: Approximately 784 people
Hope this helps!
