Question

the standard deviation in the pressure required to open a certain valve is known to be σ = 1.3 psi. due to changes in the manufacturing process, the quality - control manager feels that the pressure variability has increased
(a) state the null and alternative hypotheses in words.
(b) state the null and alternative hypotheses symbolically
(c) explain what it would mean to make a type i error
(d) explain what it would mean to make a type ii error
(a) state the null hypothesis in words. choose the correct answer below.
a. the standard deviation in the pressure required to open any valve is 1.3 psi.
b. the standard deviation in the pressure required to open a certain valve is 1.3 psi.
c. the standard deviation in the pressure required to open a certain valve is greater than 1.3 psi
d. the standard deviation in the pressure required to open a certain valve is different from 1.3 psi
state the alternative hypothesis in words. choose the correct answer below.
a. the standard deviation in the pressure required to open a certain valve is different from 1.3 psi.
b. the standard deviation in the pressure required to open a certain valve is 1.3 psi.
c. the standard deviation in the pressure required to open any valve is 1.3 psi.
d. the standard deviation in the pressure required to open a certain valve is greater than 1.3 psi

Explanation:

• Null Hypothesis: The null hypothesis (\(H_0\)) is a statement of no change or no difference. In the context of standard - deviation (a measure of variability), it assumes that the variability (standard deviation) has remained the same as the known value.
• Alternative Hypothesis: The alternative hypothesis (\(H_1\)) is the claim we are trying to find evidence for. Since the quality - control manager feels that the pressure variability has increased, the alternative hypothesis will be that the standard deviation is greater than the known value.

Answer:

• (a) State the null hypothesis in words:
• The null hypothesis is a statement about the population parameter (in this case, the standard deviation of the pressure required to open a certain valve). The known standard deviation is \(\sigma = 1.3\) psi. The null hypothesis assumes no change in the standard deviation.
• Option C: "The standard deviation in the pressure required to open a certain valve is \(1.3\) psi" is correct. Option A is wrong because it says "any valve" instead of "a certain valve" (the population under study). Option B is incorrect for the same reason as A. Option D is the alternative hypothesis for a two - tailed test (not our case here as we are dealing with an increase in variability).
• State the alternative hypothesis in words:
• Since the manager believes the pressure variability (standard deviation) has increased, the alternative hypothesis should reflect that the standard deviation is greater than \(1.3\) psi.
• Option D: "The standard deviation in the pressure required to open a certain valve is greater than \(1.3\) psi" is correct. Option A is for a two - tailed test (not relevant here). Option B is the null hypothesis. Option C is incorrect as it refers to "any valve" instead of "a certain valve".