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?

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".

[tex]\bar X=16[/tex] represent the sample mean for the sample

[tex]\mu[/tex] population mean (variable of interest)

[tex]\sigma=4[/tex] 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:

[tex]\bar X \pm z_{\alpha/2}\frac{\sigma}{\sqrt{n}}[/tex] (1)

Since the Confidence is 0.95 or 95%, the value of [tex]\alpha=0.05[/tex] and [tex]\alpha/2 =0.025[/tex], 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 [tex]z_{\alpha/2}=1.96[/tex]

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

[tex]16-1.96\frac{4}{\sqrt{25}}=14.43[/tex]

[tex]16-1.96\frac{4}{\sqrt{25}}=17.57[/tex]

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 $

akiran007 akiran007 · Answer 2 · 2020-03-12T02:30:15+01:00

Answer:

( 14.43 , 17.57 ) correctly interprets this interval.

Step-by-step 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