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Suppose that the ages of members in a large billiards league have a known standard deviation of σ = 12 σ=12sigma, equals, 12 years. Hernando plans on taking a random sample of n nn members from this population to make a 95 % 95%95, percent confidence interval for the mean age in the league. He wants the margin of error to be no more than 5 55 years. Which of these is the smallest approximate sample size required to obtain the desired margin of error?

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Answer:

[tex]n\geq 23[/tex]

Step-by-step explanation:

-For a known standard deviation, the sample size for a desired margin of error is calculated using the formula:

[tex]n\geq (\frac{z\sigma}{ME})^2[/tex]

Where:

  • [tex]\sigma[/tex] is the standard deviation
  • [tex]ME[/tex] is the desired margin of error.

We substitute our given values to calculate the sample size:

[tex]n\geq (\frac{z\sigma}{ME})^2\\\\\geq (\frac{1.96\times 12}{5})^2\\\\\geq 22.13\approx23[/tex]

Hence, the smallest desired sample size is 23

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