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Answer :
The probability that a randomly selected proofreader's age will be between 34 and 38.2 years is approximately 0.510036.
The given problem involves a normally distributed variable, where we are asked to find the probability of a proofreader's age falling between 34 and 38.2 years. We can solve this using the standard normal distribution.
To find the probability, we need to standardize the values using the formula:
[tex]$$Z = \frac{X - \mu}{\sigma}$$[/tex]
Where:
- [tex]\(X\)[/tex] is the value (age) we want to find the probability for (34 and 38.2 in this case)
- [tex]\(\mu\)[/tex] is the mean age of proofreaders (35.6 years)
- [tex]\(\sigma\)[/tex] is the standard deviation (3 years)
Calculating for \(Z\) for both values:
For [tex]\(X = 34\):[/tex]
[tex]$$Z_1 = \frac{34 - 35.6}{3} = -0.5333$$[/tex]
For [tex]\(X = 38.2\):[/tex]
[tex]$$Z_2 = \frac{38.2 - 35.6}{3} = 0.8667$$[/tex]
Now, we can use a standard normal distribution table or a calculator to find the cumulative probability associated with these [tex]\(Z\)[/tex] values. Subtracting the cumulative probability for[tex]\(Z_1\)[/tex] from the cumulative probability for [tex]\(Z_2\)[/tex] will give us the probability that the age falls between 34 and 38.2.
Using a calculator or statistical software, the cumulative probability for [tex](Z_1\ )[/tex] is approximately 0.2969 and for [tex]\(Z_2\)[/tex] is approximately 0.8069. Subtracting [tex]\(0.2969\) from \(0.8069\)[/tex] gives us the final probability of approximately [tex]\(0.510036\).[/tex]
This means there's about a 51.00% chance that a randomly selected proofreader's age will be between 34 and 38.2 years.
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