In: Statistics and Probability
A machine adjusted to fill 330 ml of beverage has a normal
distribution of 330 ml and a standard deviation of 5 ml, which is
the average amount of beverage filled into bottles. Amount of
beverage filled,
i) If it is below 326 ml, with a probability of 0.03,
ii) 0.002 probability between 326 ml and 332 ml
iii) over 332 ml with a probability of 0.10
the machine will give an error signal.
a) What is the probability that the machine gives an error
signal?
b) What is the probability that the machine has filled more than
332 ml of drinks when it is known that it gives a false
signal?
c) 8 bottles are selected regardless of these drinks. What is the
probability that more than 2 bottles are filled under 325 ml?
d) What is the probability that the average of the beverage amount
will be more than 331 ml for another sample of 36 bottles randomly
selected from these drinks?
The probability that the amount of beverage filled will be below 326 ml is
The probability that the amount of beverage filled will be over 332 ml is
The probability that the amount of beverage will be between 326 and 332 ml is
= 1 - (0.212 + 0.345) = 1 - 0.557 = 0.443
The conditional distribution table is as follows
Amount of beverage | Error Signal | No error signal | Total |
<326 | 0.03 * 0.212 = 0.00636 | 0.20564 | 0.212 |
326-332 | 0.002 * 0.443 = 0.000886 | 0.442114 | 0.443 |
>332 | 0.1 * 0.345 = 0.0345 | 0.3105 | 0.345 |
Total | 0.041746 | 0.958254 | 1 |
a) The probability that the machine gives an error signal
= 0.041746
b) The probability that the machine has filled more than 332ml of drinks given that the machine has given false signal
= 0.0345/0.041746 = 0.826
c) The probability that a bottle is filed less than 325ml is
The probability that more than 2 bottles are filled under 325 ml
d) We want to know the probability that on average, 36 randomly chosen bottles will have more than 331ml in them.
The standard error of the sample mean is
The required probability is
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