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 ​(1equalsnot ​attractive; 10equalsextremely ​attractive). Construct a confidence interval using a 99​% confidence level. What do the results tell about the mean attractiveness ratings of the population of all adult​ females? 6​, 9​, 3​, 9​, 6​, 6​, 7​, 7​, 8​, 9​, 3​, 8

Answer 1

The 99% confidence interval is be given by (4.872;8.628)

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:

`\bar X= \sum_(i=1)^n (x_i)/(n)`

The value obtained is
`\bar X=6.75`

`s=\sqrt{(\sum_(i=1)^n (x_i-\bar X))/(n-1)}`

The sample deviation obtained is
`s=2.094`

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
`\alpha=1-0.99=0.01` and
`\alpha/2 =0.005`. The degrees of freedom are given by:

`df=n-1=12-1=11`

We can find the critical values in excel using the following formulas:

"=T.INV(0.005,11)" for
`t_(\alpha/2)=-3.106`

"=T.INV(1-0.005,11)" for
`t_(1-\alpha/2)=3.106`

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

`\bar X \pm t_(\alpha/2)(s)/(√(n))`

The next step would be calculate the limits for the interval

Lower interval :

`\bar X - t_(\alpha/2)(s)/(√(n))`

`6.75 - 3.106 (2.094)/(√(12))=4.872`

Upper interval :

`6.75 + 3.106 (2.094)/(√(12))=8.628`

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.