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 your 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
I posted it once. But thanks.
To estimate the true population mean gasoline tax with 90% confidence, we can calculate a confidence interval using the given data.
First, let's calculate the sample mean and standard deviation of the gasoline taxes:
Sample Mean (x̄) = (38.4 + 40.9 + 67 + 32.5 + 51.5 + 43.4 + 38 + 43.4 + 50.7 + 35.4 + 39.3 + 41.4) / 12 = 45.6 cents
Next, we calculate the sample standard deviation:
Sample Standard Deviation (s) = √[( (38.4 - 45.6)² + (40.9 - 45.6)² + ... + (41.4 - 45.6)² ) / (12 - 1)] ≈ 9.75 cents (rounded to two decimal places)
We are looking to estimate the true population mean gasoline tax with 90% confidence. Since we have a sample size of 12, we can use a t-distribution.
Next, we need to find the t-value with a degree of freedom (df) of 11 (12 - 1) and a confidence level of 90%. Looking up the t-value in a t-table or using a statistical software, we find a t-value of approximately 1.796.
Now, we can calculate the margin of error:
Margin of Error = t-value * (s / √n)
= 1.796 * (9.75 / √12)
≈ 5.23 cents (rounded to two decimal places)
Finally, we can calculate the confidence interval:
Confidence Interval = (x̄ - Margin of Error, x̄ + Margin of Error)
= (45.6 - 5.23, 45.6 + 5.23)
≈ (40.37, 50.83) cents (rounded to two decimal places)
Based on the calculation, the 90% confidence interval for the true population mean gasoline tax is approximately (40.37, 50.83) cents.
Since the interval does not contain the national average of 44.7 cents, we can conclude that based on this sample, the national average is not likely to be 44.7 cents.