The diameter of bolts produced by a certain machine distributed with a mean of 0.30 inches and a standard deviation of 0.01 inches. What percentage of bolts will have a diameter greater than 0.32 inches.Does your answer make sense based on the given information? Explain why or why not.
notice that the z score for the given data is
(.32 - .30)/.01 = 2
using the normal distribution , I get
.0228 or 2.28%
makes sense to me.
"Explain why or why not" ?????
Makes sense since we did the math correctly
To find the percentage of bolts that will have a diameter greater than 0.32 inches, we can use the z-score formula. The z-score measures how many standard deviations a value is away from the mean.
First, let's calculate the z-score:
z = (x - μ) / σ
Where:
x = 0.32 inches (the diameter)
μ = 0.30 inches (the mean diameter)
σ = 0.01 inches (the standard deviation)
Substituting the values into the formula:
z = (0.32 - 0.30) / 0.01
z = 0.02 / 0.01
z = 2
Now, let's find the percentage of bolts that will have a diameter greater than 0.32 inches. We can use a z-table or a calculator to find the corresponding area under the normal distribution curve.
Looking up a z-score of 2 in a standard normal distribution table, we can find that the area to the left of z = 2 is approximately 0.9772.
However, we are interested in finding the percentage of bolts with a diameter greater than 0.32 inches. Hence, we subtract this value from 1:
1 - 0.9772 = 0.0228
So, approximately 2.28% of bolts will have a diameter greater than 0.32 inches.
Does the answer make sense based on the given information?
Yes, the answer makes sense based on the given information because we have a normal distribution with a mean of 0.30 inches and a standard deviation of 0.01 inches. This means that most of the bolts will have a diameter close to the mean (0.30 inches), and as we move further from the mean, the number of bolts with greater diameters will decrease. Hence, the percentage of bolts with a diameter greater than 0.32 inches is expected to be relatively low, which is confirmed by the calculated value of approximately 2.28%.
To find the percentage of bolts that will have a diameter greater than 0.32 inches, we need to use the normal distribution curve.
Step 1: Standardize the value
We will use the Z-score formula:
Z = (x - μ) / σ
where x is the value we want to find the probability for, μ is the mean, and σ is the standard deviation.
In this case, x = 0.32 inches, μ = 0.30 inches, and σ = 0.01 inches.
So, Z = (0.32 - 0.30) / 0.01 = 0.02 / 0.01 = 2.
Step 2: Look up the probability
Now that we have the Z-score, we can look up the probability corresponding to it on the standard normal distribution table. The table gives the area under the curve to the left of a particular Z-score.
Looking up a Z-score of 2 in the table, we find that the area to the left is approximately 0.9772.
Step 3: Subtract the probability from 1
To find the probability of bolts having a diameter greater than 0.32 inches, we subtract the probability from 1, since the area to the right of the Z-score represents the desired probability:
P(X > 0.32) = 1 - 0.9772 = 0.0228.
Therefore, approximately 2.28% of bolts will have a diameter greater than 0.32 inches.
As for whether this answer makes sense based on the given information, we can see that the mean diameter is 0.30 inches, and the standard deviation is 0.01 inches. Since 0.32 inches is larger than the mean, it is expected that the percentage of bolts with a diameter greater than 0.32 inches will be relatively small, and 2.28% seems reasonable in this context.