To determine whether the manufacturer needs to recalibrate the machines, you need to set up the null and alternative hypotheses. These hypotheses are based on the claim that the manufacturer wants to test if the bolts are not 4.00 cm long.

• Null Hypothesis ($H_0$): The population mean length of the bolts is equal to 4.00 cm. Mathematically, this is represented as $H_0: \mu = 4.00$.

• Alternative Hypothesis ($H_a$): The population mean length of the bolts is not equal to 4.00 cm. This indicates a two-tailed test. Mathematically, this is represented as $H_a: \mu \neq 4.00$.

These hypotheses will help you determine if there is significant evidence to suggest that the mean length of the bolts differs from 4.00 cm, indicating a need for recalibration.