For the statement below, write the claim as a mathematical statement. State the null and alternative hypotheses and identify which represents the claim. The standard deviation of the base price of a certain type of all-terrain vehicle is at least $\$ 312$. Write the claim as a mathematical statement. A. $\sigma \neq 312$ B. $\sigma \geq 312$ C. $\sigma>312$ D. $\sigma<312$ E. $\sigma=312$ F. $\sigma \leq 312$

Step 1 ：The claim is that the standard deviation of the base price of a certain type of all-terrain vehicle is at least $312. In mathematical terms, 'at least' translates to 'greater than or equal to'. Therefore, the claim can be written as \( \sigma \geq 312 \).

Step 2 ：The null hypothesis (H0) is the statement being tested. Usually the null hypothesis is a statement of 'no effect' or 'no difference'. In this case, the null hypothesis would be the opposite of the claim, which is \( \sigma < 312 \).

Step 3 ：The alternative hypothesis (H1) is the statement we accept if the null hypothesis is concluded to be untrue. The alternative hypothesis is the original claim, which is \( \sigma \geq 312 \).

Step 4 ：So, the claim represents the alternative hypothesis.

Step 5 ：Final Answer: The claim as a mathematical statement is \( \boxed{\sigma \geq 312} \). The null hypothesis is \( \boxed{\sigma < 312} \) and the alternative hypothesis is \( \boxed{\sigma \geq 312} \). The claim represents the \( \boxed{\text{alternative hypothesis}} \).