PLEASE BE VERY SPECIFIC AND SHOW EVERY SINGLE STEP INDETAIL, SPECIALLY IF USING R PROGRAM. (I NEED MOREHELP WITH PART D AND E!!!)

A bottling company uses a machine to fill bottles with a tastybeverage. The bottles are advertised to contain 300 milliliters(ml), but in reality the amount varies according to a normaldistribution with mean μ=298ml and standard deviation σ=3 ml. (Forthis problem, we’ll assume σσ is known and carry out thecalculations accordingly).

a)What is the probability that a randomly chosen bottle containsless than 296 ml?

b)Given a simple random sample of size n=6 bottles, what is theprobability that the sample mean is less than 296 ml?

c)What is the probability that a single bottle is filled within11 ml of the true mean μ=298ml? Hint: Draw the distribution andshade in what probability you want… then convert that to a questionabout standard normals. To find the answer using a table or R, youneed to look up two values and perform a subtraction.

d)What is the probability that the mean of 10 randomlyselected bottles is within 11 ml of the mean? What about the meanof a sample of n=100 bottles?

e)If a sample of size n=50 has a sample mean of ¯x=298,should this be evidence that the filling machine is out ofcalibration? i.e., assuming the machine has a mean fill amount ofμ=300 and σ=3, what is P(¯X≤298)?