Question

suppose a randomly selected sample of n = 70 men has a mean foot length of x = 27.1 cm, and the standard deviation of the sample is 2 cm. calculate an approximate 95% confidence interval for the mean foot length of men.

Answer 1

Answer on Question #76542 – Math – Statistics and Probability

Question

Suppose a randomly selected sample of $n = 70$ men has a mean foot length of $x = 27.1$ cm, and the standard deviation of the sample is 2 cm. Calculate an approximate 95% confidence interval for the mean foot length of men.

Solution

The endpoints of the approximate 95% confidence interval are

$$\bar{X} \pm z \frac{\sigma}{\sqrt{n}},$$

where $\bar{X}$ is the sample mean, $\sigma$ is the standard deviation of the sample,

$$z = \Phi^{-1} \left(1 - \frac{\alpha}{2}\right) = \Phi^{-1} (0.975) = 1.96.$$

Here $\alpha = \frac{1 - 0.95}{2} = 1.25$, $\Phi$ is the cumulative normal distributed function.

We get that the confidence interval is

$$\left(27.1 - 1.96 \frac{2}{\sqrt{70}}, 27.1 + 1.96 \frac{2}{\sqrt{70}}\right),$$ $$(26.6316, 27.5684).$$

Answer: (26.6316, 27.5684).

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