Respuesta :
Answer:
[tex]12087.7-1.96\frac{518}{\sqrt{10}}=11766.64[/tex]
[tex]12087.7+1.96\frac{518}{\sqrt{10}}=12408.76[/tex]
So on this case the 95% confidence interval would be given by (11767;12409)
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
Data: 12390 12296 11916 11713 11936 11553 12000 12428 12354 12291
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)
In order to calculate the mean and the sample deviation we can use the following formulas:
[tex]\bar X= \sum_{i=1}^n \frac{x_i}{n}[/tex] (2)
The mean calculated for this case is [tex]\bar X=12087.7[/tex]
Since the Confidence is 0.95 or 95%, the value of [tex]\alpha=0.05[/tex] and [tex]\alpha/2 =0.025[/tex], and we can use excel, a calculator or a table to find the critical value. The excel command would be: "=-NORM.INV(0.025,0,1)".And we see that [tex]z_{\alpha/2}=1.96[/tex]
Now we have everything in order to replace into formula (1):
[tex]12087.7-1.96\frac{518}{\sqrt{10}}=11766.64[/tex]
[tex]12087.7+1.96\frac{518}{\sqrt{10}}=12408.76[/tex]
So on this case the 95% confidence interval would be given by (11767;12409)
