Answer: (a) [tex]13.046<\mu< 22.15[/tex]
(b) The best point estimate of [tex]\mu[/tex] is the sample mean [tex]=17.598[/tex]
The margin of error = 4.552
Step-by-step explanation:
Given : The confidence interval for the population mean = (13.046 , 22.15)
Sample mean : [tex]\overline{x}=17.598[/tex]
Standard deviation : [tex]\sigma= 16.01712719[/tex]
Sample size : [tex]n=50[/tex]
a. Let [tex]\mu[/tex] represents the population mean.
Then we can write the confidence interval for the population mean as :-
[tex]13.046<\mu< 22.15[/tex]
b. The best point estimate of [tex]\mu[/tex] is the sample mean [tex]=17.598[/tex]
Also, the lower limit of confidence interval can be written as
[tex]\overlien{x}-E[/tex]
i.e. [tex]\overlien{x}-E=13.046[/tex]
[tex]E=\overline{x}-13.046=17.598 -13.046=4.552[/tex]
Hence, the margin of error = 4.552
The confidence interval is the range of value of a sample that represents the population.
The given parameters are:
[tex]\mathbf{\bar x = 17.598}[/tex]
[tex]\mathbf{n = 50}[/tex]
[tex]\mathbf{CI = (13.046,22.15)}[/tex]
[tex]\mathbf{\sigma_x = 16.01712719}[/tex]
(a) Express the confidence interval in the required format
We have:
[tex]\mathbf{CI = (13.046,22.15)}[/tex]
So, the required format is:
[tex]\mathbf{L < \mu < U}[/tex]
Where L and U are the lower and the upper intervals.
So, we have:
[tex]\mathbf{13.046 < \mu < 22.15}[/tex]
(b) Estimate of [tex]\mathbf{\mu}[/tex] and the margin of error
The point estimate of the population is the sample mean
The margin of error (E) is then calculated as:
[tex]\mathbf{E = \bar x - L}[/tex]
So, we have:
[tex]\mathbf{E = 17.598 - 13.046}[/tex]
[tex]\mathbf{E = 4.552}[/tex]
Hence, the margin of error is 4.522
Read more about confidence intervals at:
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