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 fine, 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 3 of 3 : Draw a conclusion and interpret the decision. Answer 2 Points We reject the null hypothesis and conclude that there is insufficient evidence at a 0.01 level of significance that the mean length of the bolts is different from 2.00 cm and the machines need to be recalibrated. We fail to reject the null hypothesis and conclude that there is insufficient evidence at a 0.01 level of significance that the mean length of the bolts is 2.00 cm and the machines need to be recalibrated. We fail to reject the null hypothesis and conclude that there is sufficient evidence at a 0.01 level of significance that the mean length of the bolts is different from 2.00 cm and the machines need to be recalibrated. We reject the null hypothesis and conclude that there is sufficient evidence at a 0.01 level of significance that the mean length of the bolts is different from 2.00 cm and the machines need to be recalibrated.

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UpStudy AI · Accepted Answer

To determine whether the manufacturer needs to recalibrate the machines based on the sample data, we will conduct a hypothesis test. Here are the steps involved:

### Step 1: State the Hypotheses
- **Null Hypothesis ($$H_0$$)**: The mean length of the bolts is equal to 2.00 cm ($$\mu = 2.00$$).
- **Alternative Hypothesis ($$H_a$$)**: The mean length of the bolts is not equal to 2.00 cm ($$\mu \neq 2.00$$).

### Step 2: Calculate the Test Statistic
We will use the Z-test for the mean since the population standard deviation is known. The formula for the Z-test statistic is:

$$
Z = \frac{\bar{x} - \mu}{\frac{\sigma}{\sqrt{n}}}
$$

Where:
- $$\bar{x} = 2.05$$ cm (sample mean)
- $$\mu = 2.00$$ cm (population mean under the null hypothesis)
- $$\sigma = 0.29$$ cm (population standard deviation)
- $$n = 169$$ (sample size)

Now, we can calculate the Z value:

$$
Z = \frac{2.05 - 2.00}{\frac{0.29}{\sqrt{169}}}
$$

Calculating the denominator:

$$
\frac{0.29}{\sqrt{169}} = \frac{0.29}{13} \approx 0.0223076923
$$

Now substituting back into the Z formula:

$$
Z = \frac{0.05}{0.0223076923} \approx 2.243
$$

### Step 3: Determine the Critical Value and Make a Decision
For a two-tailed test at a significance level of $$0.01$$, we need to find the critical Z values. The critical Z values for a significance level of $$0.01$$ (two-tailed) are approximately $$-2.576$$ and $$2.576$$.

Now we compare the calculated Z value with the critical values:
- If $$Z < -2.576$$ or $$Z > 2.576$$, we reject the null hypothesis.
- If $$-2.576 \leq Z \leq 2.576$$, we fail to reject the null hypothesis.

Since $$Z \approx 2.243$$ is within the range $$-2.576$$ to $$2.576$$, we fail to reject the null hypothesis.

### Conclusion
We fail to reject the null hypothesis and conclude that there is insufficient evidence at a $$0.01$$ level of significance that the mean length of the bolts is different from $$2.00$$ cm, and therefore, the machines do not need to be recalibrated.

### Final Answer
**We fail to reject the null hypothesis and conclude that there is insufficient evidence at a 0.01 level of significance that the mean length of the bolts is different from 2.00 cm and the machines need to be recalibrated.**
We fail to reject the null hypothesis and conclude that there is insufficient evidence at a 0.01 level of significance that the mean length of the bolts is different from 2.00 cm and the machines need to be recalibrated.

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