Question

Supercavitation is a propulsion technology for undersea vehicles that can greatly increase their speed. It occurs above approximately 50 meters per second when the pressure drops sufficiently to allow the water to dissociate into water vapor, forming a gas bubble behind the vehicle. When the gas bubble completely encloses the vehicle, supercavitation is said to occur. Eight tests were conducted on a scale model of an undersea vehicle in a towing basin with an average observed speed of 102.2 meters per second. Assume that speed is normally distributed with a known true standard deviation of 4 meters per second. Test the hypotheses that the true mean is equal to 100 meters per second vs the alternative that the mean is higher. Show all work for this problem!

Answer 1

As we are testing whether the true mean is 100, therefore the null and the alternative hypothesis here are given as:

$H_0: \mu= 100$

$H_1: \mu > 100$

The test statistic here is computed as:

$z^* = \frac{\bar X - \mu_0}{\frac{\sigma}{\sqrt{n}}} = \frac{102.2 - 100}{\frac{4}{\sqrt{8}}} =1.5556$

As this is a one tailed test, we get the p-value from standard normal tables here as:
p = P(Z > 1.5556) = 0.0599

As the p-value here is 0.0599 > 0.05 but < 0.1

Therefore the test is significant at 10% level of significance but not at 5% level of significance.

• Therefore at 5% level of significance, we dont have sufficient evidence here that the mean is more than 100
• But at 10% level of significance, we have sufficient evidence here that the mean is more than 100