Question

The amount of beer in a bottle from the Dominion filling plant is normally distributed with at mean od 330 ml and a standard deviation of 4 ml.

a. What is the probability that a single randomly selected bottle will have less than 325 ml? If the Quality Control manager at Dominion took many 12-packs of the beer (ie samples of 12):

b. What would be the approximate shape of the distribution of the sample means?

c. What would be the mean of the distribution of the sample means?

d. What would be the standard deviation of the distribution of the sample means?

e. What is the probability that a randomly selected 12-pack of beer will have a mean amount less than 325 ml?

Answer 1

It is given that the amount of beer in a bottle follows a normal distribution with mean and ml.

a. We need to find out the probability of a single randomly selected bottle will have less than 325 ml?

ie P(X<325) in a distribution with N(330,4).

For this first we need to convert this distribution to a standard normal distrbution by alculating Z score as follows:

  

Since we know X=325, and then

  

  

=-1.25

This the Z-score corresponding to 325 ml.

We need to find the P(Z<-1.25) either from the table (Fisher & Yates) or from Excel as well.

we look at the normal probability integral from Fisher and Yates table corresponding to 1.25(equal to -1.25 since the distribution is symmetric) as 0.10565.

Therefore, the probability of a single randomly selected bottle will have less than 325 ml = 0.10565. Alternatively, we can use Norm.dist function in Excel as well.

b. The engineer is drawing a sample of 12 packs from this lot, this will follow a t-distribution. Hence, the approximate shape of the sample mean is a t-distribution with men 330 ml.

c. The mean of the distribution of sample means is the same as the population mean ie = 330 ml.

d. The standard deviation of the distribution of sample mean is where is the standard deviation of the population and n is the sample size.

e. The probability that a randomly selected 12-pack of beer will have a mean amount less than 325 ml

ie with n=12 and and ie   and we need to get the mean amount less than 325 ml will correspond to

  

=-4.330

Therefore, we need to find the P(t<-4.330). Again we can use the t-table or Excel. at 11 df.

which is 0.000597 or 0.0006.