Question

A sample of n = 25 n=25 diners at a local restaurant had a mean lunch bill of $16 with a standard deviation of σ = $ 4 σ=$4 . We obtain a 95% confidence interval as ( 14.43 , 17.57 ) (14.43,17.57) . Which choice correctly interprets this interval?

Answer 1

( 14.43 , 17.57 ) correctly interprets this interval.

Explanation:

mean = x`= $ 16

Standard deviation= S= $4

sample size= n= 25

Using the formula

x`± zₐ/₂ S/ √n

= 16 ± (1.96) 4/ √25

= 16 ± (1.96) 4/5

= 16 ± (1.96)0.8

= 16 ± (1.568)

= 16+ 1.568 , 16-1.568

= 17.568 , 14.432

From 14.43 to 17.57

( 14.43 , 17.57 ) correctly interprets this interval.

If two samples are taken and two standard deviations are considered then the answer would be completely different from 13.78 to 18.22

Answer 2

`16-1.96(4)/(√(25))=14.43`

`16-1.96(4)/(√(25))=17.57`

So on this case the 95% confidence interval would be given by (14.43;17.57)

And the best interpretation would be:

We are 95% confident that the true mean for the luch bill in the local restaurant is between 14.43$ and 17.57 $

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=16` represent the sample mean for the sample

`\mu` population mean (variable of interest)

`\sigma=4` represent the population standard deviation

n=25 represent the sample size

Solution to the problem

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

`\bar X \pm z_(\alpha/2)(\sigma)/(√(n))` (1)

Since the Confidence is 0.95 or 95%, the value of
`\alpha=0.05` and
`\alpha/2 =0.025`, and we can use excel, a calculator or a table to find the critical value. The excel command would be: "=-NORM.INV(0.025,0,1)".And we see that
`z_(\alpha/2)=1.96`

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

`16-1.96(4)/(√(25))=14.43`

`16-1.96(4)/(√(25))=17.57`

So on this case the 95% confidence interval would be given by (14.43;17.57)

And the best interpretation would be:

We are 95% confident that the true mean for the luch bill in the local restaurant is between 14.43$ and 17.57 $