A random sample of state gasoline taxes in cents is shown here for 12 states. Use the data to estimate the true population mean gasoline tax with 90% confidence. Does you interval contain the national average of 44.7 cents?

38.4 40.9 67 32.5 51.5 43.4
38 43.4 50.7 35.4 39.3 41.4

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12 To estimate the true population mean gasoline tax with 90% confidence, we can use the formula for the confidence interval of a mean.

First, we need to find the sample mean and the standard error. Here are the steps:

1. Calculate the sample mean (x-bar) by summing up all the observations and dividing by the sample size:
x-bar = (38.4 + 40.9 + 67 + 32.5 + 51.5 + 43.4 + 38 + 43.4 + 50.7 + 35.4 + 39.3 + 41.4) / 12 = 45.025.

2. Calculate the standard deviation (s) of the sample by using the formula:
s = sqrt(Σ(x - x-bar)^2 / (n - 1)),
where x is each observation, x-bar is the sample mean, and n is the sample size.

For the given data, the calculations are as follows:
(38.4 - 45.025)^2 = 45.563025
(40.9 - 45.025)^2 = 20.848025
(67 - 45.025)^2 = 482.280025
(32.5 - 45.025)^2 = 158.402025
(51.5 - 45.025)^2 = 42.430025
(43.4 - 45.025)^2 = 2.622025
(38 - 45.025)^2 = 49.075025
(43.4 - 45.025)^2 = 2.622025
(50.7 - 45.025)^2 = 32.202025
(35.4 - 45.025)^2 = 92.056025
(39.3 - 45.025)^2 = 32.010025
(41.4 - 45.025)^2 = 13.260025

Σ(x - x-bar)^2 = 961.263175
s = sqrt(961.263175 / 11) = 9.157799869.

Next, we can calculate the margin of error using the t-distribution. We need the t-value with (n-1) degrees of freedom and a 90% confidence level. For this sample size of 12, the degrees of freedom are 11.

Looking up the t-value in a t-table or using a calculator, we find that the t-value for a 90% confidence level with 11 degrees of freedom is approximately 1.796.

The margin of error (ME) can be calculated by multiplying the t-value with the standard error (s) divided by the square root of the sample size (sqrt(n)):

ME = t-value * (s / sqrt(n))
= 1.796 * (9.157799869 / sqrt(12))
= 1.796 * (9.157799869 / 3.464101616)
= 1.796 * 2.643932797
= 4.743033713.

Finally, we can construct the confidence interval by subtracting and adding the margin of error to the sample mean:

Lower bound = x-bar - ME
= 45.025 - 4.743033713
= 40.28196629.

Upper bound = x-bar + ME
= 45.025 + 4.743033713
= 49.76803371.

The 90% confidence interval for the true population mean gasoline tax is (40.28196629, 49.76803371).

To determine whether the interval contains the national average of 44.7 cents, we compare it to the calculated interval. In this case, the interval does not contain the national average of 44.7 cents because 44.7 is not within the range of (40.28196629, 49.76803371).

Formula:

CI90 = mean + or - 1.645(sd/√n)

You will need to calculate mean and standard deviation from the data given. Sample size (n) = 12.

Plug values into the formula and compute the confidence interval.

I'll let you take it from here.