A process produces cable for the local telephone company. When the process is operating correctly, cable diameter follows a normal distribution with mean 1.8 inches and standard deviation 0.03 inch. A random sample of 30 pieces of cable found diameters with a sample mean of 1.806 inches and a sample standard deviation of 0.046 inch.

a. Assuming that the population standard deviation is 0.03 inch, test, at the 10% level against a two-sided alternative, the null hypothesis that the population mean is 1.8 inches. Find also the lowest significance at which this null hypothesis can be rejected against the two-sided alternative.

b. Test, at the 10% level, the null hypothesis that the population standard deviation is 0.03 inch against the alternative that it is bigger.