Respuesta :
Answer:
i) a) Normality : We assume that the data follows approximately a normal distribution
ii) Random sample: The data comes from a random sample
iii) The sample size represent <10% of the population size
We assume that all the conditions are satisfied for this case.
b) [tex]443555-2.06\frac{195381}{\sqrt{26}}=364621.22[/tex]
[tex]443555+2.06\frac{195381}{\sqrt{26}}=522488.775[/tex]
So on this case the 95% confidence interval would be given by (364621.22;522488.775)
c) We are confident at 95% that the true mean of foreclosed homes sold's are between (364621.22;522488.775)
d) Since the lower value for the 95% confidence interval is higher than 300000 we can conclude that yes differes significantly and the true mean is different from 300000 at 5% of significance.
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=443555[/tex] represent the sample mean
[tex]\mu[/tex] population mean (variable of interest)
s=195381 represent the sample standard deviation
n=26 represent the sample size
Part a
We need some conditions:
a) Normality : We assume that the data follows approximately a normal distribution
b) Random sample: The data comes from a random sample
c) The sample size represent <10% of the population size
We assume that all the conditions are satisfied for this case.
Part b
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=26-1=25[/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 tabel to find the critical value. The excel command would be: "=-T.INV(0.025,25)".And we see that [tex]t_{\alpha/2}=2.06[/tex]
Now we have everything in order to replace into formula (1):
[tex]443555-2.06\frac{195381}{\sqrt{26}}=364621.22[/tex]
[tex]443555+2.06\frac{195381}{\sqrt{26}}=522488.775[/tex]
So on this case the 95% confidence interval would be given by (364621.22;522488.775)
Part c
We are confident at 95% that the true mean of foreclosed homes sold's are between (364621.22;522488.775)
Part d
Since the lower value for the 95% confidence interval is higher than 300000 we can conclude that yes differes significantly and the true mean is different from 300000 at 5% of significance.
