Respuesta :
Answer:
[tex]\bar X = \frac{66000+70000}{2}= 68000[/tex]
We can estimate the margin of error with this formula:
[tex] ME= \frac{Upper -Lower}{2}= \frac{70000-66000}{2}= 2000[/tex]
And the margin of error is given by:
[tex] ME = z_{\alpha/2}\frac{\sigma}{\sqrt{n}} [/tex]
And we can rewrite the margin of error like this:
[tex] ME =z_{\alpha/2}*SE[/tex]
Where [tex] SE= \frac{\sigma}{\sqrt{n}}[/tex]
For 95% of confidence the critical value is [tex]z_{\alpha/2}= \pm 1.96[/tex]
The Standard error would be:
[tex] SE= \frac{ME}{z_{\alpha/2}}= \frac{2000}{1.96}= 1020.408[/tex]
For 99% of confidence the critical value is [tex]z_{\alpha/2}= \pm 2.58[/tex]
And the new margin of error would be:
[tex] ME = 2.58* 1020.408 = 2632.653[/tex]
And then the interval would be given by:
[tex] Lower = 68000- 2632.653 = 65367.347[/tex]
[tex] Upper = 68000+ 2632.653 = 70632.653[/tex]
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".
[tex]\bar X[/tex] represent the sample mean for the sample
[tex]\mu[/tex] 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:
[tex]\bar X \pm z_{\alpha/2}\frac{\sigma}{\sqrt{n}}[/tex] (1)
The 95% confidence interval is given by (66000 , 70000)
We can estimate the mean with this formula:
[tex]\bar X = \frac{66000+70000}{2}= 68000[/tex]
We can estimate the margin of error with this formula:
[tex] ME= \frac{Upper -Lower}{2}= \frac{70000-66000}{2}= 2000[/tex]
And the margin of error is given by:
[tex] ME = z_{\alpha/2}\frac{\sigma}{\sqrt{n}} [/tex]
And we can rewrite the margin of error like this:
[tex] ME =z_{\alpha/2}*SE[/tex]
Where [tex] SE= \frac{\sigma}{\sqrt{n}}[/tex]
For 95% of confidence the critical value is [tex]z_{\alpha/2}= \pm 1.96[/tex]
The Standard error would be:
[tex] SE= \frac{ME}{z_{\alpha/2}}= \frac{2000}{1.96}= 1020.408[/tex]
For 99% of confidence the critical value is [tex]z_{\alpha/2}= \pm 2.58[/tex]
And the new margin of error would be:
[tex] ME = 2.58* 1020.408 = 2632.653[/tex]
And then the interval would be given by:
[tex] Lower = 68000- 2632.653 = 65367.347[/tex]
[tex] Upper = 68000+ 2632.653 = 70632.653[/tex]
