Question

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?

Answer 1

Answer:

15 dog heights; $n=15$

Step-by-step explanation:

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

• $\gamma$ is the confidence level

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

We are given that:

• $\gamma=0.95$

• $MOE_\gamma=1$
• $z_\gamma=invNorm(0.975,0,1)=1.96$
• $\sigma=3.7$

To determine the minimum sample size, $n$, we plug our given values into the formula and solve for

$MOE_\gamma=z_\gamma*\sqrt{\frac{\sigma}{n} }$

$1=1.96\sqrt{\frac{3.7}{n} }$

$\frac{1}{1.96}=\sqrt{\frac{3.7}{n} }$

$(\frac{1}{1.96}) ^{2}=\frac{3.7}{n}$

$n=\frac{3.7}{(\frac{1}{1.96})^{2} }$

$n=14.21392$

Don't forget to round up here! This means that $n=15$ 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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