the amount of pyridoxine in grams per multiple vitamin is normally distributed with mean = 110 grams and standard deviation = 25 grams. What is the probability that a randomly selected vitamin will contain between 100 and 110 grams of pyridoxine?
(100-110)/25 = z = -.4
(110-110)/25 = z = 0
you want to find the probability between -.4 and 0
you can look this up using a z-table or a calculator.
I go .155 See if you agree.
To find the probability that a randomly selected vitamin will contain between 100 and 110 grams of pyridoxine, we need to calculate the cumulative probability at both ends of the range and subtract the two probabilities.
Step 1: Standardize the values
First, we need to standardize the values of 100 and 110 grams using the z-score formula:
z1 = (X1 - mean) / standard deviation = (100 - 110) / 25 = -0.4
z2 = (X2 - mean) / standard deviation = (110 - 110) / 25 = 0
Step 2: Find the cumulative probabilities
Next, we need to find the cumulative probabilities using the z-scores. We can use a standard normal distribution table or a calculator to find these probabilities.
P(z < -0.4) = 0.3446 (approximately)
P(z < 0) = 0.5
Step 3: Calculate the probability
Now, we can subtract the two probabilities to find the probability that a randomly selected vitamin will contain between 100 and 110 grams of pyridoxine.
P(100 ≤ X ≤ 110) = P(z1 ≤ z ≤ z2) = P(z ≤ z2) - P(z ≤ z1)
P(100 ≤ X ≤ 110) = 0.5 - 0.3446
P(100 ≤ X ≤ 110) ≈ 0.1554
Therefore, the probability that a randomly selected vitamin will contain between 100 and 110 grams of pyridoxine is approximately 0.1554 (or 15.54%).
To find the probability that a randomly selected vitamin will contain between 100 and 110 grams of pyridoxine, we'll use the concept of standard deviation and the properties of the normal distribution.
1. Calculate the z-scores for the given values:
- For the lower limit (100 grams):
z1 = (100 - mean) / standard deviation
- For the upper limit (110 grams):
z2 = (110 - mean) / standard deviation
2. Look up the corresponding probabilities from the standard normal distribution table for z1 and z2. These probabilities represent the area under the curve from negative infinity up to the respective z-scores.
3. The probability of the desired event is the difference between the probability associated with z2 and z1. This represents the area between the two z-scores under the curve, which corresponds to the probability that a randomly selected vitamin will contain between 100 and 110 grams of pyridoxine.
To calculate the z-scores, use the formula:
z = (X - mean) / standard deviation
Where:
- X is the value you want to calculate the z-score for (100 or 110 in this case).
- mean is the mean value of the distribution (110 grams).
- standard deviation is the standard deviation of the distribution (25 grams).
Using the values given, we can calculate the z-scores:
For the lower limit:
z1 = (100 - 110) / 25 = -0.4
For the upper limit:
z2 = (110 - 110) / 25 = 0
Next, look up the corresponding probabilities associated with z1 and z2 from the standard normal distribution table.
The table will provide the area under the curve up to each z-score.
The probability associated with z1 represents the area to the left of z1, while the probability associated with z2 represents the area to the left of z2.
Once you have these probabilities, subtract the probability associated with z1 from the probability associated with z2 to find the probability between the two limits.
Now, using the standard normal distribution table or a calculator, you can find the probabilities for z1 and z2:
- The probability associated with z1 is P(Z < -0.4)
- The probability associated with z2 is P(Z < 0)
From the table, the probability associated with z1 is approximately 0.3446.
The probability associated with z2 is approximately 0.5 (since P(Z < 0) is 0.5).
Finally, calculate the probability between the two limits:
P(100 < X < 110) = P(Z < 0) - P(Z < -0.4)
= 0.5 - 0.3446
= 0.1554
Therefore, the probability that a randomly selected vitamin will contain between 100 and 110 grams of pyridoxine is approximately 0.1554 (or 15.54%).