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Vikentia [17]
3 years ago
15

A survey organization has used the methods of our class to construct an approximate 95% confidence interval for the mean annual

income of households in a county. The interval runs from $66,000 to $70,000 . If possible, find an approximate 99% confidence interval for the mean annual income of households in the county. If this is not possible, explain why not.
Mathematics
1 answer:
lawyer [7]3 years ago
7 0

Answer:

\bar X = \frac{66000+70000}{2}= 68000

We can estimate the margin of error with this formula:

ME= \frac{Upper -Lower}{2}= \frac{70000-66000}{2}= 2000

And the margin of error is given by:

ME = z_{\alpha/2}\frac{\sigma}{\sqrt{n}}

And we can rewrite the margin of error like this:

ME =z_{\alpha/2}*SE

Where SE= \frac{\sigma}{\sqrt{n}}

For 95% of confidence the critical value is z_{\alpha/2}= \pm 1.96

The Standard error would be:

SE= \frac{ME}{z_{\alpha/2}}= \frac{2000}{1.96}= 1020.408

For 99% of confidence the critical value is z_{\alpha/2}= \pm 2.58

And the new margin of error would be:

ME = 2.58* 1020.408 = 2632.653

And then the interval would be given by:

Lower = 68000- 2632.653 = 65367.347

Upper = 68000+ 2632.653 = 70632.653

Step-by-step 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)

s represent the sample standard deviation

n represent the sample size  

Solution to the problem

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

\bar X \pm z_{\alpha/2}\frac{\sigma}{\sqrt{n}}   (1)

The 95% confidence interval is given by (66000 , 70000)

We can estimate the mean with this formula:

\bar X = \frac{66000+70000}{2}= 68000

We can estimate the margin of error with this formula:

ME= \frac{Upper -Lower}{2}= \frac{70000-66000}{2}= 2000

And the margin of error is given by:

ME = z_{\alpha/2}\frac{\sigma}{\sqrt{n}}

And we can rewrite the margin of error like this:

ME =z_{\alpha/2}*SE

Where SE= \frac{\sigma}{\sqrt{n}}

For 95% of confidence the critical value is z_{\alpha/2}= \pm 1.96

The Standard error would be:

SE= \frac{ME}{z_{\alpha/2}}= \frac{2000}{1.96}= 1020.408

For 99% of confidence the critical value is z_{\alpha/2}= \pm 2.58

And the new margin of error would be:

ME = 2.58* 1020.408 = 2632.653

And then the interval would be given by:

Lower = 68000- 2632.653 = 65367.347

Upper = 68000+ 2632.653 = 70632.653

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