If the mean value of the weight of a particular brand of dog food is 20.6 lb and the standard deviation is 1.3, assume a normal distribution and calculate the amount of product produced that falls below the lower specification value of 19.7 lb.
Z = (score-mean)/SD
Find table in the back of your statistics text labeled something like "areas under normal distribution" to find the proportion related to the Z score.
19.7-20.6/1.3 = -.69 = 25% or 25 percent are less than 19.7 pounds
To calculate the amount of product that falls below the lower specification value of 19.7 lb, we need to find the area to the left of this value on the normal distribution curve.
Using the Z-score formula, we can calculate the Z-score:
Z = (X - μ) / σ
where
X = lower specification value = 19.7 lb
μ = mean value = 20.6 lb
σ = standard deviation = 1.3 lb
Z = (19.7 - 20.6) / 1.3
Z = -0.692
Next, we will use a standard normal distribution table or calculator to find the corresponding area under the curve for this Z-score.
Looking up the Z-score of -0.692 in the table or using a calculator, the corresponding area is approximately 0.2454.
This means that the probability of a randomly selected product weighing less than 19.7 lb is 0.2454.
To calculate the amount of product produced that falls below this value, we need to multiply the probability by the total amount of product produced.
For example, if 1,000 lbs of product are produced:
Amount of product = 0.2454 * 1000
Amount of product = 245.4 lbs
Therefore, approximately 245.4 lbs of product will fall below the lower specification value of 19.7 lb.
To calculate the amount of product produced that falls below the lower specification value, we can use the standard normal distribution.
First, we need to standardize the lower specification value using the formula for standardizing a value in a normal distribution:
Z = (X - μ) / σ
Where:
Z is the standardized score
X is the lower specification value (19.7 lb)
μ is the mean value (20.6 lb)
σ is the standard deviation (1.3)
Plugging in the values, we get:
Z = (19.7 - 20.6) / 1.3
Z = -0.69
Next, we need to find the corresponding area under the standard normal distribution curve that corresponds to this standardized score. This area represents the probability of a randomly selected value from this distribution being less than the lower specification value.
We can use a standard normal distribution table or a calculator that calculates cumulative probabilities to find this area.
Using a standard normal distribution table, we can find the cumulative probability for Z = -0.69, which is approximately 0.2441. This means there is a 0.2441 probability that a randomly selected value from this distribution will be less than the lower specification value.
Finally, to calculate the amount of product produced that falls below the lower specification value, we multiply this probability by the total amount of product produced.
Let's assume the total amount of product produced is 1000 pounds:
Amount below lower specification value = 0.2441 * 1000
Amount below lower specification value ≈ 244.1 pounds
Therefore, approximately 244.1 pounds of product would fall below the lower specification value of 19.7 pounds.