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.)

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[/tex] represent the sample mean for the sample

[tex]\mu[/tex] 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:

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

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

[tex]\bar X= \sum_{i=1}^n \frac{x_i}{n}[/tex] (2)

[tex]s=\sqrt{\frac{\sum_{i=1}^n (x_i-\bar X)}{n-1}}[/tex] (3)

The mean calculated for this case is [tex]\bar X=2.618[/tex]

The sample deviation calculated [tex]s=0.078[/tex]

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

[tex]df=n-1=15-1=14[/tex]

Since the Confidence is 0.99 or 99%, the value of [tex]\alpha=0.01[/tex] and [tex]\alpha/2 =0.005[/tex], 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 [tex]t_{\alpha/2}=2.98[/tex]

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

[tex]2.618-2.98\frac{0.078}{\sqrt{15}}=2.56[/tex]

[tex]2.618+2.98\frac{0.078}{\sqrt{15}}=2.68[/tex]

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