Respuesta :
Answer:
a) The 95% confidence interval would be given by (29.639;31.361 )
And the results are not very different from 28.9 hg< μ < 31.9 hg
b)The 95% confidence interval would be given by (28.939;31.861)
And the results are not very different from 28.9 hg< μ < 31.9 hg
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=30.5[/tex] represent the sample mean for the sample
[tex]\mu[/tex] population mean (variable of interest)
s=6.7 represent the sample standard deviation
n=235 represent the sample size
2) First confidence interval
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=235-1=234[/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(0.025,234)".And we see that [tex]t_{\alpha/2}=1.97[/tex]
Now we have everything in order to replace into formula (1):
[tex]30.5-1.97\frac{6.7}{\sqrt{235}}=29.639[/tex]
[tex]30.5+1.97\frac{6.7}{\sqrt{235}}=31.361[/tex]
So on this case the 95% confidence interval would be given by (29.639;31.361 )
And the results are not very different from 28.9 hg< μ < 31.9 hg
3) Second confidence interval
[tex]\bar X=30.4[/tex] represent the sample mean for the sample
[tex]\mu[/tex] population mean (variable of interest)
s=2.3 represent the sample standard deviation
n=12 represent the sample size
[tex]df=n-1=12-1=11[/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(0.025,11)".And we see that [tex]t_{\alpha/2}=2.20[/tex]
Now we have everything in order to replace into formula (1):
[tex]30.4-2.20\frac{2.3}{\sqrt{12}}=28.939[/tex]
[tex]30.4+2.20\frac{2.3}{\sqrt{12}}=31.861[/tex]
So on this case the 95% confidence interval would be given by (28.939;31.861)
And the results are not very different from 28.9 hg< μ < 31.9 hg
