Question

An automobile club reported that the average price of regular gasoline in a certain state was ​$2.61 per gallon. The following data show the price per gallon of regular gasoline for 15 randomly selected stations in the state. Complete parts a through c below. 2.69 2.51 2.54 2.72 2.55 2.57 2.59 2.73 2.67 2.72 2.66 2.59 2.66 2.52 2.55 a. Construct a 99​% confidence interval to estimate the average price per gallon of gasoline in the state. The 99​% confidence interval to estimate the average price per gallon of gasoline in the state is from ​$ nothing to ​$ nothing. ​(Round to the nearest​ cent.)

Answer 1

`2.618-2.98(0.078)/(√(15))=2.56`

`2.618+2.98(0.078)/(√(15))=2.68`

So on this case the 99% confidence interval would be given by (2.56;2.68)

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)(s)/(√(n))` (1)

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

`\bar X= \sum_(i=1)^n (x_i)/(n)` (2)

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

The mean calculated for this case is
`\bar X=2.618`

The sample deviation calculated
`s=0.078`

In order to calculate the critical value
`t_(\alpha/2)` we need to find first the degrees of freedom, given by:

`df=n-1=15-1=14`

Since the Confidence is 0.99 or 99%, the value of
`\alpha=0.01` and
`\alpha/2 =0.005`, and we can use excel, a calculator or a table to find the critical value. The excel command would be: "=-T.INV(0.005,14)".And we see that
`t_(\alpha/2)=2.98`

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

`2.618-2.98(0.078)/(√(15))=2.56`

`2.618+2.98(0.078)/(√(15))=2.68`

So on this case the 99% confidence interval would be given by (2.56;2.68)