Respuesta :
Answer:
[tex]ME= \frac{4}{2}= 2[/tex]
And the margin of error is given by:
[tex] ME= t_{\alpha/2}\frac{s}{\sqrt{n}}[/tex]
[tex] s= \frac{ME \sqrt{n}}{t_{\alpha/2}}= \frac{2* \sqrt{25}}{2.0639}= 4.845[/tex]
And the sample variance is
[tex]s^2 = 4.845^2 \approx 23.47[/tex]
So then the correct option would be:
(e) The sample variance is 23.47.
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=2.5[/tex] represent the sample mean
[tex]\mu[/tex] population mean (variable of interest)
s represent the sample standard deviation
n=25 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)
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=25-1=24[/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: "=-T.INV(0.025,24)".And we see that [tex]t_{\alpha/2}=2.0639[/tex]
The lenght of the interval is given by:
[tex] 2ME = 4[/tex]
[tex]ME= \frac{4}{2}= 2[/tex]
And the margin of error is given by:
[tex] ME= t_{\alpha/2}\frac{s}{\sqrt{n}}[/tex]
And solving for the deviation we got:
[tex] s= \frac{ME \sqrt{n}}{t_{\alpha/2}}= \frac{2* \sqrt{25}}{2.0639}= 4.845[/tex]
And the sample variance is
[tex]s^2 = 4.845^2 \approx 23.47[/tex]
So then the correct option would be:
(e) The sample variance is 23.47.
