The time taken by a worker to complete a task was recorded for a random sample of 50 workers. The sample mean was 41.2 minutes and an unbiased estimate of the population variance was 32.6 minutes \({ } ^ { 2 }\). Find a \(95 \%\) confidence interval for the mean time taken to complete the task.
The probability that an \(\alpha \%\) confidence interval includes only values that are lower than the population mean is \(\frac { 1 } { 16 }\). Find the value of \(\alpha\).