Answer:
The 95% confidence interval would be given by (14444.04;33657.30)
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".
Data: $26,650 $6,060 $52,820 $8,490 $13,660$25,840 $49,840 $23,790 $51,480 $18,960$990 $11,450 $41,810 $21,060 $7,860
We can calculate the mean and the deviation from these data with the following formulas:
![\bar X= \frac{\sum_{i=1}^n x_i}{n}](https://tex.z-dn.net/?f=%5Cbar%20X%3D%20%5Cfrac%7B%5Csum_%7Bi%3D1%7D%5En%20x_i%7D%7Bn%7D)
![s=\sqrt{\frac{\sum_{i=1}^n (x_i -\bar X)^2}{n-1}}](https://tex.z-dn.net/?f=s%3D%5Csqrt%7B%5Cfrac%7B%5Csum_%7Bi%3D1%7D%5En%20%28x_i%20-%5Cbar%20X%29%5E2%7D%7Bn-1%7D%7D)
represent the sample mean for the sample
population mean (variable of interest)
s=17386.13 represent the sample standard deviation
n=15 represent the sample size
The confidence interval for the mean is given by the following formula:
(1)
In order to calculate the critical value
we need to find first the degrees of freedom, given by:
![df=n-1=15-1=14](https://tex.z-dn.net/?f=df%3Dn-1%3D15-1%3D14)
Since the Confidence is 0.95 or 95%, the value of
and
, and we can use excel, a calculator or a tabel to find the critical value. The excel command would be: "=-T.INV(0.025,14)".And we see that ![t_{\alpha/2}=2.14](https://tex.z-dn.net/?f=t_%7B%5Calpha%2F2%7D%3D2.14)
Now we have everything in order to replace into formula (1):
![24050.67+2.14\frac{17386.13}{\sqrt{15}}=33657.30](https://tex.z-dn.net/?f=24050.67%2B2.14%5Cfrac%7B17386.13%7D%7B%5Csqrt%7B15%7D%7D%3D33657.30)
So on this case the 95% confidence interval would be given by (14444.04;33657.30)