Questions: State Gasoline Taxes A random sample of state gasoline taxes (in cents) is shown here for 8 states. Round sample statistics and final answers to at least two decimal places. 32.1, 33.3, 50, 43, 44.9, 33.7, 62.8, 49.8 Use the data to estimate the true population mean gasoline tax with 80% confidence. Assume the variable is normally distributed. <μ<

Solution Steps

The mean gasoline tax is calculated using the formula: \[ \mu = \frac{\sum_{i=1}^N x_i}{N} = \frac{349.6}{8} = 43.7 \]

The variance is calculated using the formula: \[ \sigma^2 = \frac{\sum (x_i - \mu)^2}{n-1} = 112.34 \] The standard deviation is then obtained by taking the square root of the variance: \[ \sigma = \sqrt{112.34} = 10.6 \]

For an 80% confidence level, the Z-score is approximately: \[ Z = 1.28 \]

The margin of error is calculated using the formula: \[ \text{Margin of Error} = \frac{Z \times \sigma}{\sqrt{n}} = \frac{1.28 \times 10.6}{\sqrt{8}} = 4.8 \]

The 80% confidence interval for the true population mean is given by: \[ \text{Confidence Interval: } \left( \mu - \text{Margin of Error}, \mu + \text{Margin of Error} \right) = \left( 43.7 - 4.8, 43.7 + 4.8 \right) = (38.9, 48.5) \]

Final Answer