a drinks machine dispenses coffee into cups. A sign on the machine indicates that each cup contains 50 ml of coffee. the machine actually dispenses a mean amount of 55ml per cup and 10% of the cups contain less than the amount stated on the sign. assuming that the amount of coffee dispensed into each cup is normally distributed find the standard deviation of the amount of coffee dispensed per cup in ml.
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a drinks machine dispenses coffee into cups. a sign on the machine indicates that each cup contains 50 ml of coffee. the machine actually dispenses a mean amount of 55 ml per cup and 10% of the cup contain less than the amount stated on the sign. assuming that the amount of coffee dispensed into each cup is normally distributed, find (i) the standard deviation of the amount of coffee dispensed per cup in ml. (ii) the percentage of cups that contain more than 51 ml. following complaints, the owners of the machine made adjustments. only 2.5 % of the cups now contain less than 50 ml. the standard deviation of the amount dispensed is reduced to 3 ml. assuming that the amount of coffee dispensed is still normally distributed, (iii) find the new mean amount of coffee per cup. answers
To find the standard deviation of the amount of coffee dispensed per cup, we need to use the information given and the properties of the normal distribution.
First, let's calculate the z-score for the 10th percentile (10% of cups contain less than the stated amount):
z = invNorm(0.10) = -1.2816
Next, we can use the z-score formula to find the corresponding value in the distribution:
z = (X - μ) / σ
where X is the value, μ is the mean (55 ml in this case), and σ is the standard deviation (which we need to find).
Rearranging the formula to solve for σ:
σ = (X - μ) / z
Substituting the known values:
σ = (50 - 55) / -1.2816
σ = -5 / -1.2816
σ ≈ 3.90
Therefore, the standard deviation of the amount of coffee dispensed per cup is approximately 3.90 ml.
To find the standard deviation of the amount of coffee dispensed per cup, we need to use the information given regarding the mean and the percentage of cups containing less than the stated amount.
Step 1: Calculate the Z-score for the given percentage.
Since we know that 10% of cups contain less than the stated amount, we can calculate the Z-score for this percentage using a standard normal distribution table. The Z-score represents the number of standard deviations away from the mean.
From the standard normal distribution table, we find that the Z-score for a cumulative probability of 0.10 is approximately -1.28.
Step 2: Calculate the difference between the actual mean and the stated amount.
The given problem states that the mean amount of coffee dispensed per cup is 55 ml, while the stated amount is 50 ml. The difference between these two values is 55 - 50 = 5 ml.
Step 3: Calculate the standard deviation.
The standard deviation represents the spread or variability of the data. In a normal distribution, approximately 68% of the data falls within one standard deviation of the mean.
Since we know the Z-score for the cumulative probability of 0.10 is -1.28 and we have the difference of 5 ml, we can use the formula:
Z = (x - mean) / standard deviation
Substituting the values, we have:
-1.28 = (5) / standard deviation
Solving for the standard deviation:
standard deviation = 5 / (-1.28) ≈ -3.90625 ≈ 3.91 ml
Considering that standard deviation cannot be negative, we take the absolute value, giving us:
standard deviation ≈ 3.91 ml
Therefore, the standard deviation of the amount of coffee dispensed per cup is approximately 3.91 ml.