\begin{tabular}{l} A manufacturer must test that his bolts are 2.00 cm long when they come off the assembly line. He must recalibrate his machines if the bolts are too \ long or too short. After sampling 169 randomly selected bolts off the assembly line, he calculates the sample mean to be 2.05 cm . He knows that the \ population standard deviation is 0.29 cm . Assuming a level of significance of 0.01 , is there sufficient evidence to show that the manufacturer needs \ to recalibrate the machines? \ Step 1 of 3: State the null and alternative hypotheses for the test. Fill in the blank below. \ $$ \qquad H_{0}: \mu=2.00 $$ \ \hline\end{tabular}

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To solve this problem, we need to state the null and alternative hypotheses for the test regarding the length of the bolts.

### Step 1: State the Hypotheses

1. **Null Hypothesis ($$H_0$$)**: This hypothesis states that there is no effect or no difference. In this case, it asserts that the mean length of the bolts is equal to the target length. Therefore, we have:
   $$
   H_{0}: \mu = 2.00
   $$

2. **Alternative Hypothesis ($$H_a$$)**: This hypothesis states that there is an effect or a difference. In this case, it asserts that the mean length of the bolts is not equal to the target length, indicating that the manufacturer may need to recalibrate the machines. Therefore, we have:
   $$
   H_{a}: \mu \neq 2.00
   $$

### Summary of Hypotheses
- $$H_{0}: \mu = 2.00$$
- $$H_{a}: \mu \neq 2.00$$

This sets up a two-tailed test since we are checking for deviations in both directions (either too long or too short).

Next, we can proceed to the next steps of the hypothesis testing process, which typically involves calculating the test statistic and comparing it to the critical value or using a p-value approach. Would you like to continue with that?
The hypotheses are: - $$ H_{0}: \mu = 2.00 $$ - $$ H_{a}: \mu \neq 2.00 $$

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