Respuesta :
Answer:
a) The 86.64% confidence interval is given by (77.0;83.0)
b) The 97.86% confidence interval is given by (75.4;84.6)
c) The one side upper confidence interval is (80,82.2)
d) The one side lower confidence interval is (79,80)
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=80[/tex] represent the sample mean
[tex]\mu[/tex] population mean (variable of interest)
[tex]\sigma=14[/tex] represent the population standard deviation
n=49 represent the sample size
The confidence interval for the mean is given by the following formula:
[tex]\bar X \pm z_{\alpha/2}\frac{\sigma}{\sqrt{n}}[/tex] (1)
a) Develop a 86.64% confidence interval about the mean.
Since the confidence is 0.8664 or 86.64%, the value of [tex]\alpha=1-0.8664=0.1336[/tex] and [tex]\alpha/2 =0.0668[/tex], and we can use excel, a calculator or a table to find the critical value. The excel command would be: "=-NORM.INV(0.0668,0,1)".And we see that [tex]z_{\alpha/2}=1.50[/tex]
Now we have everything in order to replace into formula (1):
[tex]80-1.50\frac{14}{\sqrt{49}}=77.0[/tex]
[tex]80+1.50\frac{14}{\sqrt{49}}=83.0[/tex]
So on this case the 86.64% confidence interval would be given by (77.0;83.0)
b) Develop a 97.86% confidence interval about the mean.
Since the confidence is 0.9786 or 97.86%, the value of [tex]\alpha=1-0.9786=0.0214[/tex] and [tex]\alpha/2 =0.0107[/tex], and we can use excel, a calculator or a table to find the critical value. The excel command would be: "=-NORM.INV(0.0107,0,1)".And we see that [tex]z_{\alpha/2}=2.30[/tex]
Now we have everything in order to replace into formula (1):
[tex]80-2.30\frac{14}{\sqrt{49}}=75.4[/tex]
[tex]80+2.30\frac{14}{\sqrt{49}}=84.6[/tex]
So on this case the 97.86% confidence interval would be given by (75.4;84.6)
c) What is the upper limit for a lower one-sided 86.43% confidence interval?
Since the confidence is 0.8643 or 86.43%, the value of [tex]\alpha=1-0.8643=0.1357[/tex], and we can use excel, a calculator or a table to find the critical value. The excel command would be: "=-NORM.INV(0.1357,0,1)".And we see that [tex]z_{\alpha/2}=1.0998[/tex]
Now we have everything in order to replace into formula (1):
[tex]80+1.0998\frac{14}{\sqrt{49}}=82.2[/tex]
So the one side upper confidence interval is (80,82.2)
d) What is the lower limit for an upper one-sided 69.15% confidence interval?
Since the confidence is 0.6915 or 69.15%, the value of [tex]\alpha=1-0.6915=0.3085[/tex], and we can use excel, a calculator or a table to find the critical value. The excel command would be: "=NORM.INV(0.3085,0,1)".And we see that [tex]z_{\alpha/2}=-0.5[/tex]
Now we have everything in order to replace into formula (1):
[tex]80-0.5\frac{14}{\sqrt{49}}=79[/tex]
So the one side lower confidence interval is (79,80)
