A leading magazine (like Barron's) reported at one time that the average number of weeks an individual is unemployed is 30 weeks. Assume that for the population of all unemployed individuals the population mean length of unemployment is 30 weeks and that the population standard deviation is 2.8 weeks. Suppose you would like to select a random sample of 56 unemployed individuals for a follow-up study. Required:
a. Find the probability that a single randomly selected value is greater than 27.6.
b. Find the probability that a sample of size n=80 is randomly selected with a mean greater than 27.6.

a) The probability that a single randomly selected value is greater than 27.6

P( X≥27.6) = 0.8051

b) The probability that a  sample of size randomly selected value is greater mean greater than 27.6

P( X≥27.6) = 0.9998

Step-by-step explanation:

Step(i):-

Given that the mean of the Population = 30 weeks

The Standard deviation of the Population = 2.8 weeks

Let 'X' be a Normal distribution

The probability that a single randomly selected value is greater than 27.6

P( X≥27.6) = P(Z≥-0.857)

                 =  0.5 + A(-0.857)

                = 0.5 + A(0.857)

               = 0.5 + 0.3051

               = 0.8051

Step(ii):-

Let 'X' be a Normal distribution

The probability that a  sample of size randomly selected value is greater mean greater than 27.6

P( X≥27.6) = P(Z≥-7667)

                 = 0.5 + A(-7.667)

                 =0.5 +0.4998

                 = 0.9998

not only in mathematics, scientific proof also are required in the other field of science. also maths proofs are of particular structure, containing steps and logics. so proving is like doing science, it is science. that's why it is important to prove. 

an inaccurate proof leads to errors, or stating results that are not accurate.  which wil mess the probelm up you dont want this