The weights of boxes of Brand Z cereal were found to be normally distrubted with a mean of 16.5 ounces and a standard deviation of 0.4 ounces.

a. what percentage of the boxes will weigh more than 16 ounces?

b. what percentage of the boxes will weigh between 15.5 and 16.5 ounces?

c. if a store has 500 boxes of cereal, how many of them will weigh less than 15.5 ounces?

Z = (score-mean)/SD

Find table in the back of your statistics text labeled something like "areas under normal distribution" to find the proportions related to the Z scores.

To find the answers to these questions, we can use the standard normal distribution table or a statistical calculator. However, I will explain how to calculate these probabilities using the properties of the normal distribution.

a. To calculate the percentage of boxes that weigh more than 16 ounces, we need to find the area under the normal curve to the right of 16. We will calculate the z-score and then use the z-table or a calculator.

Step 1: Calculate the z-score
z = (x - μ) / σ
where x is the value we want to find the area for, μ is the mean, and σ is the standard deviation.

In this case, x = 16 ounces, μ = 16.5 ounces, and σ = 0.4 ounces.

z = (16 - 16.5) / 0.4
z = -1.25

Step 2: Find the area under the curve to the right of the z-score
Since we want the percentage of boxes that weigh more than 16 ounces, we need to find the complement of the area to the left of the z-score.

P(X > 16) = 1 - P(X <= 16)
Now, we can use a z-table or a calculator to find the corresponding probability:

P(X > 16) ≈ 1 - 0.8944 ≈ 0.1056
Or approximately 10.56%.

b. To calculate the percentage of boxes that weigh between 15.5 and 16.5 ounces, we need to find the area under the normal curve between these two values.

Step 1: Calculate the z-scores for both values.
For 15.5 ounces:
z1 = (15.5 - 16.5) / 0.4
z1 = -2.5

For 16.5 ounces:
z2 = (16.5 - 16.5) / 0.4
z2 = 0

Step 2: Find the area between the z-scores.
P(15.5 < X < 16.5) = P(X < 16.5) - P(X < 15.5)
Using the z-table or a calculator, we can find the corresponding probabilities:

P(X < 16.5) ≈ 0.8944
P(X < 15.5) ≈ 0.0062

P(15.5 < X < 16.5) ≈ 0.8944 - 0.0062 ≈ 0.8882
Or approximately 88.82%.

c. To calculate how many of the 500 boxes will weigh less than 15.5 ounces, we need to find the corresponding z-score and then use the z-table or a calculator.

Step 1: Calculate the z-score for 15.5 ounces.
z = (15.5 - 16.5) / 0.4
z = -2.5

Step 2: Find the area to the left of the z-score.
P(X < 15.5) ≈ 0.0062 (from the previous calculation)

Step 3: Calculate the number of boxes.
Number of boxes = P(X < 15.5) * 500
Number of boxes ≈ 0.0062 * 500 ≈ 3.1
Or approximately 3 boxes.

Therefore, approximately 3 out of the 500 boxes will weigh less than 15.5 ounces.