Two programming languages, L1 and L2, are designed to implement stochastic algorithms, which are the methods to solve computational problems by random numbers. The execution time for an instance of a standard problem is also random with the following information:

• L1: follows the normal distribution with mean 93 s and standard deviation 25 s;
• L2: follows the normal distribution with mean 100 s and standard deviation 20 s.

Though the mean of the execution time of L1 is shorter but it generates more errors due to some design flaws. The mean rates are 10 errors per 50 s of execution and 4 errors per 50 s of execution for L1 and L2 respectively.

A web server provides an interface for the public to solve this standard problem. It connects to many computational servers. It is known that 70% of the computational servers are implemented by L1 and 30% by L2. If the execution time of an instance of this problem is less than 100 s, it is evaluated as "fast".

Q7. After L1 has been redesigned and implemented, a sample of 20 instances are executed. It is found that 15 instances are "fast". The sample mean and the sample standard deviation of the execution time are 85.8 s and 39.1 s respectively. Answer the following questions based on these statistics only.

• (a) Construct a 95% confidence interval for the mean of the execution time.
• (b) Construct a 99% confidence interval for the variance of the execution time.
• (c) Without any prior assumption, determine the smallest sample size that would give a margin of error at most 0.08 for a 95% confidence interval of the proportion "fast" instances.