. The time spent by students working on a project is a normal random variable with parameters μ= 12 and ???? = 4. a. what is the probability that the amount of the time spent on a project is less than 14 hours? b. what is the probability that the amount of the time spent on a project is greater than 8 hours?

Answer:a) The probability is 0.19146.b) The probability is 0.15866.Step-by-step explanation:a) In this problem we have a random variable X with normal distribution, and its parameters are μ = 12 and σ = 4. The variable X stands for the amount of time that the students spend in a project. With this, our problem is to find the probability that X is less than 14. So,[tex]P(X\leq 14)[/tex].Recall that the tables we have are made to calculate probabilities with standardized normal variables, this means that its mean is 0 and its standard deviation is 1. This can be done considering the variable [tex]Z=\frac{x-\mu}{\sigma}. So,[tex]P(X\leq 14) = P\left(\frac{Z-12}{4}\leq \frac{14-12}{4}\right) = P(Z\leq 0.5)[/tex].Now, we look down the rows to find 0.5 and then across the columns to 0.00 which yield a probability of 0.19146.b) In this case we want to calculate [tex]P(X\geq 8)[/tex]. We follow an analogue reasoning: [tex]P(X\geq 8) = P\left \frac{Z-12}{4}\geq \frac{8-12}{4}\right) = P(Z\geq -1)[/tex]Now we use that [tex] P(Y\geq a)=1-P(Y\leq a)[/tex] where Y is a random variable. Then, [tex]P(X\geq 8) =P(Z\geq -1) = 1 -P(Z\leq -1)[/tex].But,[tex]P(Z\leq -1) = P(Z\geq 1) = 1-P(Z\leq 1)[/tex].Now we substitute this last value and get [tex]P(X\geq 8) =P(Z\geq -1) = 1 -P(Z\leq -1) = 1 - (1-P(Z\leq 1)) = P(Z\leq 1)[/tex].Again, we look into the table and found that [tex]P(X\geq 8) =P(Z\leq 1) = 0.15866[/tex].