Respuesta :
Answer:
(8.510, 8.610)
Step-by-step explanation:
1) 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=8.56[/tex] represent the sample mean
[tex]\mu[/tex] population mean (variable of interest)
s=0.09 represent the sample standard deviation
n=15 represent the sample size
2) Confidence interval
The confidence interval for the mean is given by the following formula:
[tex]\bar X \pm t_{\alpha/2}\frac{s}{\sqrt{n}}[/tex] (1)
In order to calculate the critical value [tex]t_{\alpha/2}[/tex] we need to find first the degrees of freedom, given by:
[tex]df=n-1=15-1=14[/tex]
Since the Confidence is 0.98 or 98%, the value of [tex]\alpha=0.02[/tex] and [tex]\alpha/2 =0.01[/tex], and we can use excel, a calculator or a table to find the critical value. The excel command would be: "=-T.INV(0.01,14)".And we see that [tex]t_{\alpha/2}=2.62449[/tex]
Now we have everything in order to replace into formula (1):
[tex]8.56-2.62449\frac{0.09}{\sqrt{15}}=8.500[/tex]
[tex]8.56-2.62449\frac{0.09}{\sqrt{15}}=8.621[/tex]
So on this case the 95% confidence interval would be given by (8.500;8.621)
If we assume that the population deviation [tex]\sigma=0.09[/tex] then the confidence interval is given by:
[tex]\bar X \pm z_{\alpha/2}\frac{\sigma}{\sqrt{n}}[/tex] (2)
Since the Confidence is 0.98 or 98%, the value of [tex]\alpha=0.02[/tex] and [tex]\alpha/2 =0.01[/tex], and we can use excel, a calculator or a table to find the critical value. The excel command would be: "=-NORM.INV(0.01,1,0)".And we see that [tex]z_{\alpha/2}=2.33[/tex]
Now we have everything in order to replace into formula (2):
[tex]8.56-2.33\frac{0.09}{\sqrt{15}}=8.510[/tex]
[tex]8.56-2.33\frac{0.09}{\sqrt{15}}=8.614[/tex]
So on this case the 95% confidence interval would be given by (8.510;8.610)
Based on this the most accurate answer would be:
(8.510, 8.610)
