Use the table to answer each question. Note: Round z-scores to the nearest hundredth and then find the required A values using the table.
The weights of all the boxes of corn flakes filled by a machine are normally distributed, with a mean weight of 13.0 ounces and a standard deviation of 0.4 ounce. What percent of the boxes will have the following weights? (Round your answers to one decimal place.)
(a) weigh less than 12.5 ounces
%
(b) weigh between 12 ounces and 14 ounces
To answer these questions using the table, we need to find the corresponding z-scores and then look up the corresponding areas under the standard normal curve.
(a) To find the percentage of boxes that weigh less than 12.5 ounces, we need to find the area to the left of the z-score for 12.5.
First, we calculate the z-score using the formula:
z = (x - μ) / σ
Where x is the value we want to find the proportion for, μ is the mean, and σ is the standard deviation.
In this case,
x = 12.5 ounces
μ = 13.0 ounces
σ = 0.4 ounce
Substituting these values into the formula:
z = (12.5 - 13.0)/0.4
z = -0.5/0.4
z = -1.25
Next, we need to find the area to the left of this z-score using the standard normal table.
Looking up -1.25 in the table gives us a value of 0.1056.
To find the percentage, we multiply this value by 100:
0.1056 * 100 = 10.56%
So, approximately 10.6% of the boxes will weigh less than 12.5 ounces.
(b) To find the percentage of boxes that weigh between 12 ounces and 14 ounces, we need to find the area between the z-scores for these weights.
First, we calculate the z-scores for each weight:
For 12 ounces:
z = (12 - 13.0)/0.4
z = -1/0.4
z = -2.5
For 14 ounces:
z = (14 - 13.0)/0.4
z = 1/0.4
z = 2.5
Next, we need to find the area between these z-scores.
Looking up -2.5 in the table gives us a value of 0.0062.
Looking up 2.5 in the table gives us a value of 0.9938.
To find the area between these z-scores, we subtract the smaller value from the larger value:
0.9938 - 0.0062 = 0.9876
To find the percentage, we multiply this value by 100:
0.9876 * 100 = 98.76%
So, approximately 98.8% of the boxes will weigh between 12 ounces and 14 ounces.