The Department of Transportation in a foreign county establishes gas-mileage standards that automobiles sold in must meet or else a "gas guzzler" tax is imposed on the sale of the offending types of automobile. For the "compact, four-door" class of automobiles, the target average gas mileage is 25 miles per gallon.

Achievement of the standard is tested by randomly choosing 20 cars from a manufacturer's assembly line, and then examining the distance between the vector of 20 observed measurements of gas mileage/gallon and a \((20 \times 1)\) vector of targeted gas mileages for these cars. Letting \(\mathbf{X}\) represent the \(20 \times 1\) vector of observed gas mileages, and letting \(\mathbf{t}\) represent the \(20 \times 1\) vector of targeted gas mileages (i.e., \(\mathbf{t}=(25,25, \ldots, 25)^{\prime}\) ), the distance measure is \(D(\mathbf{x}, \mathbf{t})=\left[(\mathbf{x}-\mathbf{t})^{\prime}(\mathbf{x}-\mathbf{t})ight]^{1 / 2}\).

If \(D(\mathbf{x}, \mathbf{t}) \leq 6\), then the type of automobile being tested is judged to be consistent with the standard; otherwise, the type of automobile will be taxed.

Specific Motors Company is introducing a new four-door compact into the market, and has requested that this type of automobile be tested for adherence to the gas-mileage standard. The engineers at Specific Motors know that the miles per gallon achieved by their compact four-door automobile can be represented by a normal distribution with mean 25 and variance 1.267, so that the target gas mileage is achieved on average.

(a) What is the probability that a car randomly chosen from Specific Motor's assembly line will be within 1 mile per gallon of the gas-mileage standard?

(b) What is the probability that Specific Motor's compact four-door will be judged as being consistent with the gas-mileage standard?

(c) A neighboring country uses a simpler test for determining whether the gas-mileage standard is met. It also has a target of 25 miles per gallon, but its test involves forming the simple average of the 20 randomly observed miles per gallon, and then simply testing whether the calculated average is within one mile per gallon of 25 miles per gallon. That is, the gasmileage standard will be judged to have been met if

\[\frac{1}{20} \sum_{i=1}^{20} x_{i} \in[24,26] .\]

What is the probability that Specific Motors will pass this alternative test?