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The population standard deviation for the heights of dogs, in inches, in a city is 3.7 inches. If we want to be 95% confident that the sample mean is within 1 inch of the true population mean, what is the minimum sample size that can be taken?

Respuesta :

Answer:

15 dog heights; [tex]n=15[/tex]

Step-by-step explanation:

The formula to be used here is [tex]MOE_\gamma=z_\gamma*\sqrt{\frac{\sigma}{n} }[/tex] where:

  • [tex]\gamma[/tex] is the confidence level
  • [tex]MOE_\gamma[/tex] is the margin of error for a confidence level
  • [tex]z_\gamma[/tex] is the critical value for the confidence level
  • [tex]\sigma[/tex] is the population standard deviation
  • [tex]n[/tex] is the sample size

We are given that:

  • [tex]\gamma=0.95[/tex]
  • [tex]MOE_\gamma=1[/tex]
  • [tex]z_\gamma=invNorm(0.975,0,1)=1.96[/tex]
  • [tex]\sigma=3.7[/tex]

To determine the minimum sample size, [tex]n[/tex], we plug our given values into the formula and solve for

[tex]MOE_\gamma=z_\gamma*\sqrt{\frac{\sigma}{n} }[/tex]

[tex]1=1.96\sqrt{\frac{3.7}{n} }[/tex]

[tex]\frac{1}{1.96}=\sqrt{\frac{3.7}{n} }[/tex]

[tex](\frac{1}{1.96}) ^{2}=\frac{3.7}{n}[/tex]

[tex]n=\frac{3.7}{(\frac{1}{1.96})^{2} }[/tex]

[tex]n=14.21392[/tex]

Don't forget to round up here! This means that [tex]n=15[/tex] actually.

Therefore, if we want to be 95% confident that the sample mean is within 1 inch of the true population mean, the minimum sample size that can be taken is 15 dog heights.

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