Question

1. The Glen Valley Steel Company manufacturers steel bars. If the production process is working properly, it turns out steel bars with mean length of a least 2.8 feet with a standard deviation of 0.20 foot (as determined from engineering specifications on the production equipment involved). Longer steel bars can be used or altered, but shorter bars must be scrapped. You select a sample of 25 bars and the mean length is 2.73 feet. Do you need to adjust the production equipment?
   1. If you want to test the null hypothesis at the 0.05 level of significance, what decision would you make using the critical value approach to hypothesis testing?
   2. If you want to test the null hypothesis at the 0.05 level of significance, what decision would you make using the p-value approach to hypothesis testing?
   3. Interpret the meaning of the p-value in this problem.

Answer 1

(a) The hypothesis being tested is:

H0: µ 2.8

Ha: µ < 2.8

z = (x - µ)/σ/√n

z = (2.73 - 2.8)/0.20/√25

z = -1.75

The critical z-value is -1.96.

Since -1.75 > -1.96, we can reject the null hypothesis.

Therefore, we can conclude that we need to adjust the production equipment.

(b) The hypothesis being tested is:

H0: µ 2.8

Ha: µ < 2.8

z = (x - µ)/σ/√n

z = (2.73 - 2.8)/0.20/√25

z = -1.75

The p-value for z = -1.75 is 0.0401.

Since the p-value (0.0401) is smaller than the significance level (0.05), we can reject the null hypothesis.

Therefore, we can conclude that we need to adjust the production equipment.

(c) The p-value gives the probability of rejecting the null hypothesis.

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