Question

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

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

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

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

c)What is the probability that a single bottle is filled within 11 ml of the true mean μ=298ml? Hint: Draw the distribution and shade in what probability you want… then convert that to a question about standard normals. To find the answer using a table or R, you need to look up two values and perform a subtraction.

d)What is the probability that the mean of 10 randomly selected bottles is within 11 ml of the mean? What about the mean of 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 of calibration? i.e., assuming the machine has a mean fill amount of μ=300 and σ=3, what is P(¯X≤298)?

Answer 1

a) Let X denotes the amount of a randomly chosen bottle.

b) Let denotes the mean amount for random sample of 6 bottles.

c) X ~ Normal(298, 3²)

The probability that a single bottle is filled within 11 ml of the true mean μ=298ml

d) or

Now,

The probability that the mean of 10 randomly selected bottles is within 11 ml of the mean

For a sample of size 100,

or

Now,

The probability that the mean of 100 randomly selected bottles is within 11 ml of the mean

e)