Part a). We asked to calculate "P(X > 10)" . Let's change the variable
Then "Z" will be distributed according to standard normal distribution "Z \\sim N(0,1)". Let's rewrite "P(X > 10)" as
(at the last one we use simmetry of the standard normal distribution about "x = 0"). Now let's use the following table
and get "P(Z < \\frac{1}{2}) \\approx 0.6915", thus
"P(X > 10) \\approx 0.6915"Part b). We asked to calculate "P(13 < X < 14)". Using the same variable "Z" we can rewrite it as
Using the table "P(Z < \\frac{1}{2}) \\approx 0.6915" and "P(Z < \\frac{1}{4}) \\approx 0.5987" . Thus
Part c). Let "{X_i}" be the amount of ounces of soda that the i-th customer will get, and "{X_i} \\sim N(12,4)" . Let's use the following lemma: Let "\\xi \\sim N({\\mu _1},{\\sigma _1})" and "\\eta \\sim N({\\mu _2},{\\sigma _2})" be the two normally distributed random values. Then "(\\xi + \\eta ) \\sim N({\\mu _1} + {\\mu _2},\\sqrt {\\sigma _1^2 + \\sigma _2^2} )" . You may find two different proofs of this here
https://en.wikipedia.org/wiki/Sum_of_normally_distributed_random_variables
Using this lemma we see that
Now we need to find "P(\\frac{Y}{{100}} < 12.24) = P(Y < 1224)" . Let's change the variable
and we get
Using the table "P(Z < \\frac{3}{5}) \\approx 0.7257" thus
This is the answer.
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