Respuesta :
Answer:
The 98% confidence interval would be given by (25.904;29.306)
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".
The data is:
21, 15, 35, 40 ,40 ,34, 32, 11, 18, 35, 9, 31, 15, 29, 41, 37, 40, 10, 31, 14, 39, 14, 11, 33, 38, 21, 35, 33, 34, 27, 31, 26, 35, 27, 32, 18, 40, 17
2) Compute the sample mean and sample standard deviation.
In order to calculate the mean and the sample deviation we need to have on mind the following formulas:
[tex]\bar X= \sum_{i=1}^n \frac{x_i}{n}[/tex]
[tex]s=\sqrt{\frac{\sum_{i=1}^n (x_i-\bar X)}{n-1}}[/tex]
=AVERAGE(21, 15, 35, 40 ,40 ,34, 32, 11, 18, 35, 9, 31, 15, 29, 41, 37, 40, 10, 31, 14, 39, 14, 11, 33, 38, 21, 35, 33, 34, 27, 31, 26, 35, 27, 32, 18, 40, 17)
On this case the average is [tex]\bar X= 27.605[/tex]
=STDEV.S(21, 15, 35, 40 ,40 ,34, 32, 11, 18, 35, 9, 31, 15, 29, 41, 37, 40, 10, 31, 14, 39, 14, 11, 33, 38, 21, 35, 33, 34, 27, 31, 26, 35, 27, 32, 18, 40, 17)
The sample standard deviation obtained was s=10.184
[tex]\sigma=4.5[/tex] represent the population standard deviation
3) Find the critical value t* Use the formula for a CI to find upper and lower endpoints
In order to find the critical value we need to take in count that our sample size n =38 >30 and on this case we know about the population standard deviation, so on this case we need to use the z distribution. Since our interval is at 98% of confidence, our significance level would be given by [tex]\alpha=1-0.98=0.02[/tex] and [tex]\alpha/2 =0.01[/tex].
We can find the critical values in excel using the following formulas:
"=NORM.INV(0.01,0,1)" for [tex]z_{\alpha/2}=-2.33[/tex]
"=T.INV(1-0.01,0,1)" for [tex]z_{1-\alpha/2}=2.33[/tex]
The confidence interval for the mean is given by the following formula:
[tex]\bar X \pm z_{\alpha/2}\frac{\sigma}{\sqrt{n}}[/tex]
And we can use Excel to calculate the limits for the interval
Lower interval : "=27.605 -2.33*(4.5/SQRT(38))" =25.904
Upper interval : "=27.605 +2.33*(4.5/SQRT(38))" =29.306
So the 98% confidence interval would be given by (25.904;29.306)
