(Solved): 1) The average and standard deviation of thickness of a steel plate from 25 samples are measured as ...

1) The average and standard deviation of thickness of a steel plate from 25 samples are measured as \( 33 \mathrm{~mm} \) and \( 3 \mathrm{~mm} \), respectively. (13 marks) a) If the actual standard deviation thickness is \( 2 \mathrm{~mm} \), determine the probability of getting a sample standard deviation result as high or higher than \( 3 \mathrm{~mm} \). Write necessary assumption(s). b) What does your answer of part a imply about the true standard deviation of the thickness being \( 2 \mathrm{~mm} \) ? c) How large of a sample is needed to be \( 96 \% \) confident that the sample mean will be within \( 1 \mathrm{~mm} \) of the actual mean, if the actual standard deviation is later confirmed to be \( 2.5 \mathrm{~mm} \) ?