Respuesta :
Answer:
The 80% confidence interval for the mean per capita income in thousands of dollars is between $21.3 and $21.9.
Step-by-step explanation:
We have that to find our [tex]\alpha[/tex] level, that is the subtraction of 1 by the confidence interval divided by 2. So:
[tex]\alpha = \frac{1-0.8}{2} = 0.1[/tex]
Now, we have to find z in the Ztable as such z has a pvalue of [tex]1-\alpha[/tex].
So it is z with a pvalue of [tex]1-0.1 = 0.9[/tex], so [tex]z = 1.28[/tex]
Now, find the margin of error M as such
[tex]M = z*\frac{\sigma}{\sqrt{n}}[/tex]
In which [tex]\sigma[/tex] is the standard deviation of the population and n is the size of the sample.
[tex]M = 1.28*\frac{12.5}{\sqrt{3839}} = 0.3[/tex]
The lower end of the interval is the sample mean subtracted by M. So it is 21.6 - 0.3 = $21.3.
The upper end of the interval is the sample mean added to M. So it is 21.6 + 0.3 = $21.9.
The 80% confidence interval for the mean per capita income in thousands of dollars is between $21.3 and $21.9.
Answer:
[tex]21.6-1.28\frac{12.5}{\sqrt{3839}}=21.3[/tex]
[tex]21.6+1.28\frac{12.5}{\sqrt{3839}}=21.9[/tex]
So on this case the 80% confidence interval would be given by (21.3;21.9)
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=21.6[/tex] represent the sample mean for the sample
[tex]\mu[/tex] population mean (variable of interest)
[tex]\sigma=12.5[/tex] represent the population standard deviation
n=3839 represent the sample size
Solutio n 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)
Since the Confidence is 0.80 or 80%, the value of [tex]\alpha=0.2[/tex] and [tex]\alpha/2 =0.1[/tex], and we can use excel, a calculator or a table to find the critical value. The excel command would be: "=-NORM.INV(0.1,0,1)".And we see that [tex]z_{\alpha/2}=1.28[/tex]
Now we have everything in order to replace into formula (1):
[tex]21.6-1.28\frac{12.5}{\sqrt{3839}}=21.3[/tex]
[tex]21.6+1.28\frac{12.5}{\sqrt{3839}}=21.9[/tex]
So on this case the 80% confidence interval would be given by (21.3;21.9)
