The age distribution of a sample of 5000 people is bell-shaped with a mean of 40 years and a standard deviation of 12 years. Determine the approximate percentage of people who are 16 to 64 years old.
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To determine the approximate percentage of people who are 16 to 64 years old, we need to find the cumulative area under the curve between those two ages.
First, we need to standardize the ages of 16 and 64.
For 16 years old:
z = (x - μ) / σ = (16 - 40) / 12 = -24 / 12 = -2
For 64 years old:
z = (x - μ) / σ = (64 - 40) / 12 = 24 / 12 = 2
Next, we can use a standard normal distribution table or a calculator to find the corresponding cumulative probabilities for these z-scores.
The cumulative probability for z = -2 is approximately 0.0228.
The cumulative probability for z = 2 is approximately 0.9772.
To find the percentage of people who are between 16 and 64 years old, we subtract the smaller cumulative probability from the larger one:
Percentage = (0.9772 - 0.0228) * 100 = 0.9544 * 100 = 95.44%
Therefore, approximately 95.44% of people in the sample are between 16 to 64 years old.
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