Question

1. A single serving of potato chips is considered to be about 42 grams (about 1.5 ounces). Single-serving bags are filled by a machine but because the chips vary in size and weight, the total weight in a bag varies according to a normal distribution. Suppose the machine is set so that the mean weight is 45 grams. A small proportion of the bags end up weighing less than 42 grams due to the random distribution. The manufacturer wants to REDUCE the proportion that weighs less than 42 grams.   Use a sketch to help you decide which option below will accomplish this. (Assume the standard deviation is fixed --i.e., cannot be changed.)

	a.	There is nothing they can do to reduce the proportion that ends up weighing less than 42 grams.
	b.	If they set the machine so the mean is less than 42 grams, they can reduce the proportion that ends up weighing less than 42 grams.
	c.	If they set the machine so the mean is closer to 42 grams, they can reduce the proportion that ends up weighing less than 42 grams.
	d.	If they set the machine so the mean is greater than 45 grams, they can reduce the proportion that ends up weighing less than 42 grams.

2. This question refers to the potato chip bag-filling machine problem in Question #6 above. The problem is still how to reduce the proportion of bags that are filled with less than 42 grams of chips. This time, assume the mean must stay at 45 grams, but the company can alter the standard deviation. Use a sketch to decide how the company can proceed if they want to reduce this proportion by changing the standard deviation.

	a.	No matter how we change the standard deviation, if the mean stays 45 grams, the proportion of bags weighing less than 42 grams will not change.
	b.	If they make the standard deviation larger, they can reduce the proportion of bags that are filled to less than 42 grams.
	c.	It depends what the standard deviation is now. We have to know its value before we can decide how to change it.
	d.	If they make the standard deviation smaller, they can reduce the proportion of bags that are filled to less than 42 grams.

Answer 1

Let mean be denoted by M and standard deviation by sd.

For a normal distribution 68.2% observations lie in the interval( M +sd,M-sd)

Similarly around 95.4% ovservations lie between the interval (M +2sd, M-2sd)

Simiarly for 3sd it becomes 99.8%.

So the basic property is mean is fixed and observations is clustered around the mean and reduces as we go away from the mean. Also if we reduce the standard deviation the observations become more close to the mean and more observations are included in the space around it.

sSo in the first question the sd is fixed so the only way to reduce the proportion of bags weighing less than 42 grams is to increase the mean to more than 45 grams. So option 4 is correct. The concept is clear from the plot of the normal distribution attached.

In the second question mean is fixed but we can change sd. We can achieve the result by reducing the sd so that more observations becomes centered around mean and less fall below the 42 grams as required. So option 4 is correct. This is also clear from the plot of the normal distribution.