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Braun Shedd · Answer 1 · 2021-11-09T08:37:13+0000

Answer:

The confidence interval for the mean is given by the following formula:

$\bar X \pm t_(\alpha/2)(s)/(√(n))$ (1)

The margin of error is given by this formula:

$ME=z_(\alpha/2)(\sigma)/(√(n))$ (2)

And on this case we have that ME =4 and we are interested in order to find the value of n, if we solve n from equation (2) we got:

$n=((z_(\alpha/2) \sigma)/(ME))^2$ (3)

Since the Confidence is 0.90 or 90%, the value of
$\alpha=0.1$ and
$\alpha/2 =0.005$ , and we can use excel, a calculator or a table to find the critical value. The excel command would be: "=-NORM.INV(0.005,0,1)".And we see that
$z_(\alpha/2)=1.64$

$n=((1.64(22))/(4))^2 = 81.36\approx 82$

So the answer for this case would be n=82 rounded up to the nearest integer

Explanation:

Previous concepts

A confidence interval is "a range of values that’s likely to include a population value with a certain degree of confidence. It is often expressed a % whereby a population means lies between an upper and lower interval".

The margin of error is the range of values below and above the sample statistic in a confidence interval.

Normal distribution, is a "probability distribution that is symmetric about the mean, showing that data near the mean are more frequent in occurrence than data far from the mean".

$\bar X$ represent the sample mean for the sample

$\mu$ population mean (variable of interest)

$\sigma$ represent the population standard deviation

n represent the sample size

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