according to a manufacturer, the average weight of a cereal box they produced is 20 ounces with a standard deviation of 0.5 ounce.
a) if a random sample of 1000 boxes are selected, what is the probability that the weight is less than 19.5 ounces?
b) what is the probability that the cereal boxes weigh more than 21 ounces?
c) what is the probability that the cereal boxes weigh between 19.95 and 21.5 ounces?
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To solve these questions, we need to use the Z-score formula and the Standard Normal Distribution table, also known as the Z-table.
a) To find the probability that the weight of a cereal box is less than 19.5 ounces, we first need to calculate the Z-score for this value. The Z-score is calculated using the formula:
Z = (X - μ) / σ
Where:
X = the value we want to find the probability for (in this case, 19.5 ounces)
μ = the mean weight of the cereal boxes (20 ounces in this case)
σ = the standard deviation of the cereal box weights (0.5 ounces in this case)
Substituting the values into the formula:
Z = (19.5 - 20) / 0.5
Z = -1 / 0.5
Z = -2
Now, we look up the Z-score -2 in the Z-table to find the corresponding probability. Looking in the table, we find that the probability corresponding to a Z-score of -2 is approximately 0.0228.
Therefore, the probability that the weight of a cereal box is less than 19.5 ounces is 0.0228, or 2.28%.
b) Similarly, to find the probability that the cereal boxes weigh more than 21 ounces, we calculate the Z-score:
Z = (X - μ) / σ
Substituting the values into the formula:
Z = (21 - 20) / 0.5
Z = 1 / 0.5
Z = 2
Looking up the Z-score 2 in the Z-table, we find that the probability corresponding to it is approximately 0.9772.
Therefore, the probability that the weight of a cereal box is more than 21 ounces is 0.9772, or 97.72%.
c) To find the probability that the cereal boxes weigh between 19.95 and 21.5 ounces, we need to calculate the probabilities for both of these values separately and then subtract the probability of 19.95 from the probability of 21.5.
First, calculate the Z-scores for both values:
Z1 = (19.95 - 20) / 0.5
Z1 = -0.05 / 0.5
Z1 = -0.1
Z2 = (21.5 - 20) / 0.5
Z2 = 1.5 / 0.5
Z2 = 3
Looking up these Z-scores in the Z-table, we find that the probabilities corresponding to Z1 and Z2 are approximately 0.4602 and 0.9987, respectively.
Subtracting these probabilities:
P = 0.9987 - 0.4602
P = 0.5385
Therefore, the probability that the weight of a cereal box is between 19.95 and 21.5 ounces is 0.5385, or 53.85%.