The diameters of pencils produced by a certain machine are normally distributed with a mean of 0.39 inches and a standard deviation of 0.04 inches.
If 500 pencils are selected how many would you expect to have a diameter more than 0.33 inches?
.33 = .39-.06 = .39 - .04*1.5
so, that is 1.5 std below the mean.
Now consult your Z table and find that
P(Z > -1.5) = .9332
.9332*500 = 466.6
Thank you so much. I greatly appreciate it.
To find out how many pencils you would expect to have a diameter more than 0.33 inches, you can use the standard normal distribution.
Step 1: Calculate the z-score.
The z-score is calculated using the formula:
z = (x - μ) / σ
where x is the value you're interested in, μ is the mean, and σ is the standard deviation.
In this case, the value x is 0.33 inches, the mean μ is 0.39 inches, and the standard deviation σ is 0.04 inches. Plugging in these values, we get:
z = (0.33 - 0.39) / 0.04
z = -1.5
Step 2: Calculate the probability.
Now that we have the z-score, we can use a standard normal distribution table or calculator to find the probability associated with that z-score. The probability represents the proportion of pencils with a diameter more than 0.33 inches.
Looking up a z-score of -1.5 in the standard normal distribution table or using a calculator, we find that the corresponding probability is approximately 0.0668.
Step 3: Multiply the probability by the total number of pencils.
To find out how many pencils you would expect to have a diameter more than 0.33 inches out of a total of 500 pencils, you can simply multiply the probability by 500:
expected number = probability * total number
expected number = 0.0668 * 500
expected number ≈ 33.4
Therefore, you would expect approximately 33 pencils to have a diameter more than 0.33 inches out of the 500 pencils selected.
To find out how many pencils would be expected to have a diameter more than 0.33 inches, we need to calculate the probability of a pencil having a diameter more than 0.33 inches. We can then multiply this probability by the total number of pencils (500) to get the expected number.
Let's use the standard normal distribution to find the probability. We can convert the given values to z-scores using the formula:
z = (x - μ) / σ
where:
x = 0.33 (the diameter we are interested in)
μ = 0.39 (mean diameter)
σ = 0.04 (standard deviation)
Calculating the z-score:
z = (0.33 - 0.39) / 0.04
z = -0.06 / 0.04
z = -1.5
Next, we can use a z-table or a statistical calculator to find the probability of a z-score being less than -1.5. By looking up the z-score of -1.5 in the table, we can find the corresponding probability.
Let's assume the probability is P(z < -1.5) = 0.0668 (obtained from the z-table).
To find the probability of a pencil having a diameter more than 0.33 inches, we can subtract this probability from 1 (since P(z < -1.5) gives the probability of being less than -1.5).
P(diameter > 0.33) = 1 - P(z < -1.5)
P(diameter > 0.33) = 1 - 0.0668
P(diameter > 0.33) = 0.9332
Now, we can multiply the probability by the total number of pencils to get the expected number:
Expected number = P(diameter > 0.33) * Total number of pencils
Expected number = 0.9332 * 500
Expected number ≈ 466.6
Therefore, we would expect approximately 466 pencils to have a diameter more than 0.33 inches.