Question

Before agreeing to purchase a large order of polyethylene sheaths for a particular type of high-pressure oil-filled submarine power cable, a company wants to see conclusive evidence that the true standard deviation of sheath thickness is less than 0.06 mm. What hypotheses should be tested, and why?

The appropriate hypotheses are

H0: σ ? < > ≠ = 0.06 mm

versus

Ha: σ ? < > ≠ = 0.06 mm.

With this formulation, the burden of proof is on the data to show that the requirement ---Select--- has not has been met.

In this context, what are the type I and type II errors?

In this context, the type I error occurs if we ---Select--- accept / reject a shipment that should have been ---Select--- accepted / rejected .

A type II error occurs if we ---Select--- accept / reject a shipment that should have been ---Select--- accepted / rejected .

Answer 1

Explanation:

Hello!

To get into context I'll define first the study variable.

X: Thickness of a polyethylene sheath.

The company won't purchase unless the standard deviation of sheath thickness is less than 0.06 mm, symbolically δ < 0.06

As a rule, the null hypothesis always states the "no change" case or the already known information about the study parameter, so it always carries the = symbol.

Note: Depending on the statistic course you are taking, you will be taught to do hypothesis tests for the standard deviation or that it is not valid to use it as a parameter. Personally, because δ is not directly a population parameter and, above all, because no statistic allows it to be studied directly. That is, to make a test for the standard deviation you have to use the Chi-square distribution, which includes the variance. I never use δ in the statistical hypotheses, just in case your course accept the hypothesis for the population standard deviation, here are the two options.

Hypotheses for δ

H₀: δ ≥ 0.06

H₁: δ < 0.06

Type I error:

Accept a shipment when it should have been rejected. (The standard deviation of sheath thickness is at least 0.06 mm)

Type II error:

Reject a shipment that should have been accepted. (The standard deviation of sheath thickness is less than 0.06 mm)

Hypotheses for δ₂

H₀: δ₂ ≥ 0.0036

H₁: δ₂ < 0.0036

Type I error:

Accept a shipment when it should have been rejected. (The population variance of sheath thickness is at least 0.0036 mm)

Type II error:

Reject a shipment that should have been accepted. (The population variance of sheath thickness is less than 0.0036 mm)

I hope it helps!