Question

in a study of speed​ dating, male subjects were asked to rate the attractiveness of their female​ dates, and a sample of the results is listed below ​(1equals=not ​attractive; 10equals=extremely ​attractive). Construct a confidence interval using a 9090​% confidence level. What do the results tell about the mean attractiveness ratings of the population of all adult​ females?

Answer 1

Answer:

$7-1.796\frac{2.216}{\sqrt{12}}=5.851$    

$7+1.796\frac{2.216}{\sqrt{12}}=8.149$    

So on this case the 90% confidence interval would be given by (5.851;8.149)    

Step-by-step explanation:

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".

$\bar X$ represent the sample mean for the sample  

$\mu$ population mean (variable of interest)

s represent the sample standard deviation

n represent the sample size  

Solution to the problem

The confidence interval for the mean is given by the following formula:

$\bar X \pm t_{\alpha/2}\frac{s}{\sqrt{n}}$   (1)

Data: 7 8 2 10 6 5 7 8 8 9 5 9

In order to calculate the mean and the sample deviation we can use the following formulas:  

$\bar X= \sum_{i=1}^n \frac{x_i}{n}$ (2)  

$s=\sqrt{\frac{\sum_{i=1}^n (x_i-\bar X)}{n-1}}$ (3)  

The mean calculated for this case is $\bar X=7$

The sample deviation calculated $s=2.216$

In order to calculate the critical value $t_{\alpha/2}$ we need to find first the degrees of freedom, given by:

$df=n-1=12-1=11$

Since the Confidence is 0.90 or 90%, the value of $\alpha=0.1$ and $\alpha/2 =0.05$, and we can use excel, a calculator or a table to find the critical value. The excel command would be: "=-T.INV(0.05,11)".And we see that $t_{\alpha/2}=1.796$

Now we have everything in order to replace into formula (1):

$7-1.796\frac{2.216}{\sqrt{12}}=5.851$    

$7+1.796\frac{2.216}{\sqrt{12}}=8.149$    

So on this case the 90% confidence interval would be given by (5.851;8.149)