Respuesta :
Answer:
The 99% confidence interval would be given by (98.536;98.864)
We are 99% confident that the true mean body temperature is between (98.536;98.864)
The value 98.6 is included on the interval but the mid point for the interval is the sample mean 98.7, so for this case 98.6 would be a value to high in order to estimate the population mean, since the best estimator for the population mean is the sample mean on this case [tex]\hat \mu =\bar X =98.7[/tex].
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=98.7[/tex] represent the sample mean for the sample
[tex]\mu[/tex] population mean (variable of interest)
s=0.64 represent the sample standard deviation
n=105 represent the sample size
Construct a 99% confidence interval estimate of the mean body temperature of all healthy humans.
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=105-1=104[/tex]
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(0.005,104)".And we see that [tex]t_{\alpha/2}=2.62[/tex]
Now we have everything in order to replace into formula (1):
[tex]98.7-2.62\frac{0.64}{\sqrt{105}}=98.536[/tex]
[tex]98.7+2.62\frac{0.64}{\sqrt{105}}=98.864[/tex]
So on this case the 99% confidence interval would be given by (98.536;98.864)
What does the sample suggest about the use of 98.6 degrees F as the mean body temperature?
We are 99% confident that the true mean body temperature is between (98.536;98.864)
The value 98.6 is included on the interval but the mid point for the interval is the sample mean 98.7, so for this case 98.6 would be a value to high in order to estimate the population mean, since the best estimator for the population mean is the sample mean on this case [tex]\hat \mu =\bar X =98.7[/tex].
