Respuesta :
Answer:
a) The 95% confidence interval would be given by (323.003;337.397)
b) The 99% confidence interval would be given by (320.351;340.049)
Step-by-step explanation:
1) 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=330.2[/tex] represent the sample mean for the sample
[tex]\mu[/tex] population mean (variable of interest)
s=15.4 represent the sample standard deviation
n=20 represent the sample size
2) 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=20-1=19[/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(0.025,19)".And we see that [tex]t_{\alpha/2}=2.09[/tex]
Now we have everything in order to replace into formula (1):
[tex]330.2-2.09\frac{15.4}{\sqrt{20}}=323.003[/tex]
[tex]330.2+2.09\frac{15.4}{\sqrt{20}}=337.397[/tex]
So on this case the 95% confidence interval would be given by (323.003;337.397)
3) Part b
Since the Confidence is 0.99 or 99%, the value of [tex]\alpha=0.01[/tex] and [tex]\alpha/2 =0.005[/tex], and we can use excel, a calculator or a table to find the critical value. The excel command would be: "=-T.INV(0.005,19)".And we see that [tex]t_{\alpha/2}=2.86[/tex]
Now we have everything in order to replace into formula (1):
[tex]330.2-2.86\frac{15.4}{\sqrt{20}}=320.351[/tex]
[tex]330.2+2.86\frac{15.4}{\sqrt{20}}=340.049[/tex]
So on this case the 99% confidence interval would be given by (320.351;340.049)
