Respuesta :
Answer:
a) [tex]884-2.426\frac{50}{\sqrt{40}}=864.82[/tex]
[tex]884+2.426\frac{50}{\sqrt{40}}=903.179[/tex]
So on this case the 98% confidence interval would be given by (864.82;903.179)
b) For this case the upper bound of the confidence interval is lower than 950 so we don't have a significant evidence to conclude that the true mean is 950 per month
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=884[/tex] represent the sample mean
[tex]\mu[/tex] population mean (variable of interest)
s=50 represent the sample standard deviation
n=40 represent the sample size
Solution to the problem
Part a
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=40-1=39[/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(0.01,39)".And we see that [tex]t_{\alpha/2}=2.426[/tex]
Now we have everything in order to replace into formula (1):
[tex]884-2.426\frac{50}{\sqrt{40}}=864.82[/tex]
[tex]884+2.426\frac{50}{\sqrt{40}}=903.179[/tex]
So on this case the 98% confidence interval would be given by (864.82;903.179)
Part b
For this case the upper bound of the confidence interval is lower than 950 so we don't have a significant evidence to conclude that the true mean is 950 per month
