Respuesta :
Answer:
The 99% confidence interval is be given by (4.872;8.628)
Step-by-step explanation:
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 dataset is:
6, 9, 3, 9, 6, 6, 7, 7, 8, 9, 3, 8
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]
The value obtained is [tex]\bar X=6.75[/tex]
[tex]s=\sqrt{\frac{\sum_{i=1}^n (x_i-\bar X)}{n-1}}[/tex]
The sample deviation obtained is [tex]s=2.094[/tex]
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 =12 <30 and on this case we don't know about the population standard deviation, so on this case we need to use the t distribution. Since our interval is at 99% of confidence, our significance level would be given by [tex]\alpha=1-0.99=0.01[/tex] and [tex]\alpha/2 =0.005[/tex]. The degrees of freedom are given by:
[tex]df=n-1=12-1=11[/tex]
We can find the critical values in excel using the following formulas:
"=T.INV(0.005,11)" for [tex]t_{\alpha/2}=-3.106[/tex]
"=T.INV(1-0.005,11)" for [tex]t_{1-\alpha/2}=3.106[/tex]
The confidence interval for the mean is given by the following formula:
[tex]\bar X \pm t_{\alpha/2}\frac{s}{\sqrt{n}}[/tex]
The next step would be calculate the limits for the interval
Lower interval :
[tex]\bar X - t_{\alpha/2}\frac{s}{\sqrt{n}}[/tex]
[tex]6.75 - 3.106 \frac{2.094}{\sqrt{12}}=4.872[/tex]
Upper interval :
[tex]6.75 + 3.106 \frac{2.094}{\sqrt{12}}=8.628[/tex]
So the 99% confidence interval would be given by (4.872;8.628)
99% of the time, when we calculate a confidence interval with a sample of n=12, the true mean of rate of attractiveness of their female dates will be between the 4.872 and 8.628.
