Suppose that if a signal of intensity \(\mu\) is emitted from a particular star, then the value received at an observatory on earth is a normal random variable with mean \(\mu\) and standard deviation 4. In other words, the value of the signal emitted is altered by random noise, which is normally distributed with mean 0 and standard deviation 4. It is suspected that the intensity of the signal is equal to 10.

1. Find a 95% confidence interval for \(\mu\).

2. Test whether this hypothesis is plausible if the same signal is independently received 20 times and the average of the 20 values received is 11.6. Use the 5 percent level of significance.

It is extremely important in a certain chemical process that a solution to be used as a reactant have a pH level greater than 8.40. A method for determining pH that is available for solutions of this type is known to give measurements that are normally distributed with a mean equal to the actual pH and with a standard deviation of 0.05. Suppose 10 independent measurements yielded the following pH values:

8.30, 8.42, 8.44, 8.32, 8.43, 8.41, 8.42, 8.46, 8.37, 8.42

Suppose it is a very serious mistake to run the process with a reactant having a pH level less than or equal to 8.40.

1. What null hypothesis should be tested? What is the alternative hypothesis?

2. Using the 5% level of significance, what would you advise to use or not to use the solution?

Among a clinic's patients having high blood cholesterol levels of at least 240 milliliters per deciliter of blood serum, volunteers were recruited to test a new drug designed to reduce blood cholesterol. A group of 40 volunteers were given the drug for 60 days, and the changes in their blood cholesterol levels were noted. If the average change was a decrease of 6.8 with a sample standard deviation of 12.1, what conclusions can we draw? Use the 5 percent level of significance.