a) 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 $32.6{\text{ }}minute{s^2}$. Find a 95% confidence interval for the mean time taken to complete the task.
b) 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 $.
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