Question

A soda pop company wants to ensure that each of the cans that they produce has exactly 12 ounces of fluid in it.  They go to three of their plants and measure the mean and standard deviation for 10 cans of soda systemically taken off the assembly line.  Form conclusions based on the means and sample deviations found.  

	Mean	Standard deviations
Plant A	11.5	0.01
Plant B	12.0	0.53
Plant C	12.0	0.02

1. What would be the soda pop company’s ideal mean and standard deviation?  Explain.

2. Which plant is the most consistent at filling their cans?  Why?

3. How likely is it that plant A is filling any of their cans to 12 ounces? Use the empirical Rule (68, 95, 99.7) to make justification.

d) Which of the plants is the least consistent at filling their cans?

e) In statistic’s 5% is often used to indicate unusual events.  If a distribution is normal, this 5% would give the probability of not being within 2 standard deviations of the mean. Considering plant C, how many ounces would count as an unusual event?

Answer 1

We have details of three plants of the company, and we have mean and variance of the data collected from those 3 plants

Based on this information, we can find an estimate of the true mean. We can have a combined mean of these 3 plants and use it as the ideal mean

In each of the plants, they have taken measurements on 10 soda cans.

Then, we know, if there are 3 groups with group sizes n₁ , n₂ , n₃ and i^th group mean is for i = 1,2,3

Then the combined mean will be

Here, n = 10 + 10 + 10 = 30 and

Putting the values we get ,

Also, we can find the value of pooled variance and this can be used as an ideal variance.

The formula for pooled variance is given by,

and i^th plant mean.

Here, n_{i =} 10 for all i = 1,2,3

Putting the values we get,

=0.1494 (up to 4 decimal places)

Hence Ideal standard deviation is

(Though this is an biased estimate of the population variance because we are using n as divisor in stead of using n-1 as divisor for finding the combined variance

b) Consistency is inversely proportional with variance.

Hence less variance implies better consistency.

Here plant A has minimum variance among all 3 plants.

Hence plant A is most consistent at filling their cans.

d) Clearly, from the given data we can observe that plant B has highest variation (because of highest standard deviation , which is 0.53) in filling up the cans. Hence it has the least consistency in filling up the cans.

c) Empirical rule states us that for a normal population, within 1 standard deviation away from mean 68% of the total data will fall, again, within 2 standard deviation away from mean, there will be 95% of the total data, and within 3 standard deviation away from mean, 99.7 % of the data will fall.

Here for plant a we see that the standard deviation is 0.01, hence if we consider 1 standard deviation, then we can say that inside the range (11.5-0.01,11.5+0.01) = (11.49,11.51), 68 % of the data will fall, that is 68% of the bottles will have filled up with soda in the range of amount (11.49,11.51)

If we consider 3 standard deviation away from mean then we get the range (11.47,11.53) and in this range 99.7% of the data will fall.

Hence it is clear that it is very much less likely for plant A to fill up 12 ounce in their bottles.

The probability that plant A will fill up their cans up to 12 ounce is much less than (1-0.9973) = 0.0027.

(Here all the justifications are based on the assumption that the amount in the bottles approximately follows a normal distribution)

e)

In case of plant c , 2 standard deviation away means (2*0.02) ounce = 0.04 ounces away.

Hence, the range will be (12.00-0.04,12.00+0.04)

= (11.96,12.04)

Hence any value coming outside this range will be considered as an unusual event.