Respuesta :
Answer:
[tex] 9.9676 - 2.326*0.5904 =8.594[/tex]
[tex] 9.9676 + 2.326*0.5904 =11.341[/tex]
Step-by-step explanation:
Notation
[tex]\bar X[/tex] represent the sample mean
[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 t_{\alpha/2}\frac{s}{\sqrt{n}}[/tex] (1)
For this case the 9% confidence interval is given by:
[tex] 8.8104 \leq \mu \leq 11.1248[/tex]
We can calculate the mean with the following:
[tex]\bar X = \frac{8.8104 +11.1248}{2}= 9.9676[/tex]
And we can find the margin of error with:
[tex] ME= \frac{11.1248- 8.8104}{2}= 1.1572[/tex]
The margin of error for this case is given by:
[tex] ME = t_{\alpha/2}\frac{s}{\sqrt{n}} = t_{\alpha/2} SE[/tex]
And we can solve for the standard error:
[tex] SE = \frac{ME}{t_{\alpha/2}}[/tex]
The critical value for 95% confidence using the normal standard distribution is approximately 1.96 and replacing we got:
[tex] SE = \frac{1.1572}{1.96}= 0.5904[/tex]
Now for the 98% confidence interval the significance is [tex]\alpha=1-0.98= 0.02[/tex] and [tex]\alpha/2 = 0.01[/tex] the critical value would be 2.326 and then the confidence interval would be:
[tex] 9.9676 - 2.326*0.5904 =8.594[/tex]
[tex] 9.9676 + 2.326*0.5904 =11.341[/tex]
