Answer :
Answer:
[tex] \bar X = \frac{461.2+598.6}{2}=529.9[/tex]
[tex] ME= \frac{Width}{2}= \frac{137.4}{2}= 68.7[/tex]
The margin of error is given by:
[tex] ME = t_{\alpha/2} SE[/tex]
For 95% of confidence the value of the significance is [tex] \alpha =0.05[/tex] and [tex] \alpha/2 = 0.025[/tex], the degrees of freedom are given by:
[tex] df = 12-1 = 11[/tex]
And then we can calculate the critical value for 95% with df = 11 and we got [tex] t_{\alpha}= 2.20[/tex]
And then we can find the standard error:
[tex] SE = \frac{ME}{t_{\alpha/2}}= \frac{68.7}{2.20}= 31.227[/tex]
[tex] t_{\alpha/2}= 3.11[/tex]
And using the confidence interval formula we got:
[tex] \bar X \pm t_{\alpha/2} SE[/tex]
[tex] 529.9 - 3.11*31.227 = 432.784[/tex]
[tex] 529.9 + 3.11*31.227 = 627.016[/tex]
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
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 the 95% confidence interval we know that the result is given by (461.2, 598.6), we can estimate the sample mean like this:
[tex] \bar X = \frac{461.2+598.6}{2}=529.9[/tex]
Because the distribution is symmetrical, now we can estimate the width of the interval like this:
[tex] Width = 598.6-461.2= 137.4[/tex]
And the margin of error is given by:
[tex] ME= \frac{Width}{2}= \frac{137.4}{2}= 68.7[/tex]
The margin of error is given by:
[tex] ME = t_{\alpha/2} SE[/tex]
For 95% of confidence the value of the significance is [tex] \alpha =0.05[/tex] and [tex] \alpha/2 = 0.025[/tex], the degrees of freedom are given by:
[tex] df = 12-1 = 11[/tex]
And then we can calculate the critical value for 95% with df = 11 and we got [tex] t_{\alpha}= 2.20[/tex]
And then we can find the standard error:
[tex] SE = \frac{ME}{t_{\alpha/2}}= \frac{68.7}{2.20}= 31.227[/tex]
The standard error is given by [tex] SE= \frac{s}{\sqrt{n}}[/tex]
Now we are interested for the 99% confidence interval, so then we need to find a new critical value for this confidence level and we got:
[tex] t_{\alpha/2}= 3.11[/tex]
And using the confidence interval formula we got:
[tex] \bar X \pm t_{\alpha/2} SE[/tex]
[tex] 529.9 - 3.11*31.227 = 432.784[/tex]
[tex] 529.9 + 3.11*31.227 = 627.016[/tex]