The weight of a bag of corn chips is normally distributed with a mean of 22
ounces and a standard deviation of 0.5 ounces.
-The probability that a bag of corn chips weighs more than 21 ounces is?
-The probability that a bag of corn chips weighs between 20.75 and 23.25
ounces is?
z = (21-22)/.5
z = -2
1-.0062 = .9938
z = (20.75-22//.5
z = -2.5
z =( 23.25-22)/.5
z = 2.5
.9938-.0228 = .9710
Statistics - Kuai, Sunday, October 27, 2013 at 8:55pm
z = (21-22)/.5
z = -2
1-.0228 = .9710
z = (20.75-22//.5
z = -2.5
z =( 23.25-22)/.5
z = 2.5
.9938-.0062= .9876
To find the probabilities, we can use the Z-score formula with the given mean and standard deviation.
The Z-score formula is given by:
Z = (X - μ) / σ
Where:
Z is the Z-score
X is the value we are interested in
μ is the mean
σ is the standard deviation
1. Probability that a bag of corn chips weighs more than 21 ounces:
To find this probability, we need to find the area under the curve to the right of 21 ounces.
First, we calculate the Z-score:
Z = (21 - 22) / 0.5 = -2
Using a Z-table or a statistical calculator, we can find the area to the right of -2:
P(Z > -2) = 1 - P(Z <= -2)
P(Z > -2) ≈ 1 - 0.0228
P(Z > -2) ≈ 0.9772
So, the probability that a bag of corn chips weighs more than 21 ounces is approximately 0.9772.
2. Probability that a bag of corn chips weighs between 20.75 and 23.25 ounces:
To find this probability, we need to find the area under the curve between 20.75 and 23.25 ounces.
First, we calculate the Z-scores for both values:
Z1 = (20.75 - 22) / 0.5 = -1.5
Z2 = (23.25 - 22) / 0.5 = 2.5
Using the Z-table or a statistical calculator, we can find the area to the right of -1.5 and to the left of 2.5 and subtract them to find the desired probability:
P(-1.5 < Z < 2.5) = P(Z < 2.5) - P(Z < -1.5)
P(-1.5 < Z < 2.5) ≈ 0.9938 - 0.0668
P(-1.5 < Z < 2.5) ≈ 0.9270
So, the probability that a bag of corn chips weighs between 20.75 and 23.25 ounces is approximately 0.9270.
To calculate the probabilities, we will use the standard normal distribution. However, since we are given the mean and standard deviation of the weight of the corn chips, we will have to convert the values to z-scores.
A z-score represents the number of standard deviations a particular value is from the mean. The formula to calculate the z-score is:
z = (x - μ) / σ
Where:
- z is the z-score
- x is the value we want to convert to a z-score
- μ is the mean
- σ is the standard deviation
Now, let's calculate the probabilities.
1. The probability that a bag of corn chips weighs more than 21 ounces:
To find this probability, we need to find the area under the normal curve to the right of 21 ounces.
First, calculate the z-score for x = 21:
z = (21 - 22) / 0.5 = -2
Now, we need to find the area to the right of -2 on the standard normal distribution. This can be done using a standard normal distribution table or by using statistical software.
Looking up the z-score of -2 in the standard normal distribution table or using software, we can find the probability associated with it. Let's assume the probability is P(-2) = 0.0228.
Since the area to the left of a z-score represents the cumulative probability, we need to subtract this value from 1 to find the probability to the right of 21 ounces:
Probability = 1 - P(-2) = 1 - 0.0228 = 0.9772
Therefore, the probability that a bag of corn chips weighs more than 21 ounces is approximately 0.9772.
2. The probability that a bag of corn chips weighs between 20.75 and 23.25 ounces:
To find this probability, we need to find the area under the normal curve between these two values.
First, let's calculate the z-scores for each value:
For x = 20.75:
z1 = (20.75 - 22) / 0.5 ≈ -2.5
For x = 23.25:
z2 = (23.25 - 22) / 0.5 ≈ 4.5
Now, we need to find the area between z1 and z2 on the standard normal distribution. Using the standard normal distribution table or statistical software, we can find the probabilities associated with these z-scores.
Let's assume the probability associated with z1 is P(z1) ≈ 0.0062, and the probability associated with z2 is P(z2) ≈ 1.
To find the probability between these two z-scores, we subtract the smaller probability from the larger probability:
Probability = P(z2) - P(z1) = 1 - 0.0062 ≈ 0.9938
Therefore, the probability that a bag of corn chips weighs between 20.75 and 23.25 ounces is approximately 0.9938.