Lazurus Steel Corporation produces iron rods that are supposed to be 36 inches long. The machine that makes these rods does not produce each rod exactly 36 inches long. The lengths of the rods are normally distributed and vary slightly. It is known that when the machine is working properly, the mean length of the rods is 36 inches. The standard deviation of the lengths of all rods produced on this machine is always equal to 0.035 inch. The quality control department at the company takes a sample of 20 such rods every week, calculates the mean length of these rods, and tests the null hypothesis, u = 36 inches, against the alternative hypothesis, u + 36 inches. If the null hypothesis is rejected, the machine is stopped and adjusted. A recent sample of 20 rods produced a mean length of 36.015 inches. Calculate the p-value for this test of hypothesis. Based on this p-value, will the quality control inspector decide to stop the machine and adjust it if he chooses the maximum probability of a Type I error to be 0.01? Use the normal distribution table. Round your answer to four decimal places. p-value =

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MATLAB: An Introduction with Applications

6th Edition

ISBN: 9781119256830

Author: Amos Gilat

Publisher: John Wiley & Sons Inc

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Question

**Overview**

Lazurus Steel Corporation manufactures iron rods intended to be 36 inches long. However, due to variations in the manufacturing process, the exact length of each rod may differ slightly. The lengths of these rods follow a normal distribution around a mean value, with occasional deviations. To ensure quality, the mean length when the machinery operates correctly should be 36 inches, with a standard deviation of 0.035 inch.

**Quality Control Process**

- Each week, the quality control department obtains a sample of 20 rods to assess their mean length.
- The department tests the null hypothesis \( \mu = 36 \) inches against the alternative hypothesis \( \mu \neq 36 \) inches.

**Recent Observations**

- In a recent test, the sample mean length was 36.015 inches for 20 rods.

**Instruction for Analysis**

You're tasked with calculating the \( p \)-value for this hypothesis test. If the \( p \)-value suggests significant deviation from the expected mean (with a significance level \(\alpha = 0.01\)), the machinery will be halted for adjustments.

**Calculation Guidance**

- Use the normal distribution table.
- Provide the \( p \)-value rounded to four decimal places.

**Question**

Based on your \( p \)-value, should the quality control inspector adjust the machine if the threshold for Type I error probability is 0.01?

\( p \)-value = ____

(Note: Ensure you perform the calculations needed to fill in the \( p \)-value.)

Transcribed Image Text:The manufacturer of a certain brand of auto batteries claims that the mean life of these batteries is 45 months. A consumer protection agency that wants to check this claim took a random sample of 24 such batteries and found that the mean life for this sample is 43.25 months. The lives of all such batteries have a normal distribution with the population standard deviation of 4.5 months. Find the p-value for the test of hypothesis with the alternative hypothesis that the mean life of these batteries is less than 45 months. Round your answer to four decimal places. \[ p = \]

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