The weights of ice cream cartons are normally distributed with a mean weight of 1313 ounces and a standard deviation of 0.50.5 ounce. (a) What is the probability that a randomly selected carton has a weight greater than 13.1713.17 ounces? (b) A sample of 3636 cartons is randomly selected. What is the probability that their mean weight is greater than 13.1713.17 ounces?

a) We are interested on this probability

And the best way to solve this problem is using the normal standard distribution and the z score given by:

If we apply this formula to our probability we got this:

b)

The new z score is given by:

And replacing we got:

Step-by-step explanation:

Previous concepts

Normal distribution, is a "probability distribution that is symmetric about the mean, showing that data near the mean are more frequent in occurrence than data far from the mean".

The Z-score is "a numerical measurement used in statistics of a value's relationship to the mean (average) of a group of values, measured in terms of standard deviations from the mean".  

Part a

Let X the random variable that represent the weights of a population, and for this case we know the distribution for X is given by:

Where and

We are interested on this probability

And the best way to solve this problem is using the normal standard distribution and the z score given by:

If we apply this formula to our probability we got this:

Part b

Since the distribution for X is normal then we can conclude that the distribution for the sample mean is given by:

And we want this probability:

The new z score is given by:

And replacing we got:

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step-by-step explanation: