
normal distributions
A machine is used to fill bags with beans. The machine is set to add 10 kilograms of beans to each bag. The table shows the weights of 277 bags that were randomly selected.
i.Are the weights normally distributed? How do you know?
ii.Do you think that using the machine is acceptable and fair to the customers? Explain your reasoning
wt in kg 9.5 9.6 9.7 9.8 9.9 10 10.1 10.2 10.3 10.4 10.5
#of bags 1 3 13 25 41 66 52 41 25 7 3
the mean 10
standard deviation 0.32
(meanstand.dev) 9.68 <= x<= 10.32(mean + stand. dev) to fall in the first standard deviation(68%)
8/11* 100 = 72 %(sounds good)
Is the machine acceptable to the customers?? No idea

For the first question, I would say no because the weights are not 'symmetrical' about the mean.
$\displaystyle \begin{array}{cccccccccccc} \hline
Wt. /kg & 9.5 & 9.6 & 9.7&9.8&9.9&10&10.1&10.2&10.3&10.4&10.5 \\ \hline
\text{No. of bags} & 1&2&3&25&41&66&52&41&25&7&3 \\ \hline \end{array}$
I would have said it is symmetrical if it was:
$\displaystyle \begin{array}{cccccccccccc} \hline
Wt. /kg & 9.5 & 9.6 & 9.7&9.8&9.9&10&10.1&10.2&10.3&10.4&10.5 \\ \hline
\text{No. of bags} & 1&2&3&25&41&66&41&25&3&2&1 \\ \hline \end{array}$
Or
$\displaystyle \begin{array}{cccccccccccc} \hline
Wt. /kg & 9.5 & 9.6 & 9.7&9.8&9.9&10&10.1&10.2&10.3&10.4&10.5 \\ \hline
\text{No. of bags} & 3&7&25&41&52&66&52&41&25&7&3 \\ \hline \end{array}$
The mean is 10.04 g to the 2nd decimal place. And I'm getting the standard deviation as 0.182 instead of 0.32...
But looking at the shape of distribution, I would say it is fair ecause it is negatively skewed (it is 'pushed' towards the right a little more).