For a specific type of machine part being produced, the diameter is normally distributed, with a mean of 15.000 cm and a standard deviation of 0.030 cm. Machine parts with a diameter more than 2 standard deviations away from the mean are rejected.
If 15,000 machine parts are manufactured, how many of these will be rejected?
A) 560
B) 680
C) 750
D) 970
I think its A
if mean is 15.000 and we reject anything over 2 standard deviations away, we reject anything over 15.06 and under 14.94
go to
http://davidmlane.com/hyperstat/z_table.html
(the best stats page on the net)
enter 15 for the mean
enter .03 for SD
click on between and enter
14.94 and 15.06 and click "calculate" to get .0455
.0455(15000) = 682.5
The closest choice is B) 680
You are probably using "tables" or charts from a text book. This webpage is more accurate.
To find the number of machine parts that will be rejected, we need to calculate how many standard deviations away from the mean a machine part with diameter greater than or equal to 2 standard deviations away will be.
The formula to calculate the number of standard deviations away from the mean is:
z = (x - μ) / σ
Where:
z = number of standard deviations away from the mean
x = value of the diameter
μ = mean of the diameter distribution
σ = standard deviation of the diameter distribution
In this case, we want to find the number of machine parts with a diameter greater than or equal to 2 standard deviations away from the mean. So we need to calculate:
z = (x - μ) / σ
z = (x - 15.000) / 0.030
z = 2
Now, we need to find the area under the normal distribution curve to the right of z = 2, which represents the proportion of machine parts that will be rejected.
We can use a standard normal distribution table to find this area. Looking up the value of z = 2 in the table, we find that the area to the right of z = 2 is approximately 0.0228.
Finally, to find the number of rejected machine parts, we multiply the proportion by the total number of machine parts:
Number of rejected machine parts = Proportion * Total number of machine parts
Number of rejected machine parts = 0.0228 * 15,000
Number of rejected machine parts ≈ 342
Therefore, the correct answer is not listed among the options provided.
To solve this problem, we need to calculate the z-score for the threshold value and then find the proportion of machine parts that fall beyond this threshold.
Step 1: Calculate the threshold value
A machine part will be rejected if its diameter is more than 2 standard deviations away from the mean. Since the mean is 15.000 cm and the standard deviation is 0.030 cm, the threshold value can be calculated as follows:
Threshold value = Mean + (2 * Standard Deviation)
Threshold value = 15.000 + (2 * 0.030)
Threshold value = 15.000 + 0.060
Threshold value = 15.060 cm
Step 2: Calculate the z-score
The z-score measures the number of standard deviations a data point is above or below the mean. We can calculate the z-score using the formula:
z-score = (x - Mean) / Standard Deviation
For the threshold value, the z-score can be calculated as:
z-score = (Threshold value - Mean) / Standard Deviation
z-score = (15.060 - 15.000) / 0.030
z-score = 0.060 / 0.030
z-score = 2
Step 3: Find the proportion of machine parts beyond the threshold
The proportion can be calculated using the standard normal distribution table or a statistical calculator. The proportion for a z-score of 2 is approximately 0.0228.
Step 4: Calculate the number of rejected machine parts
To find the number of rejected machine parts, we multiply the proportion by the total number of machine parts manufactured:
Number of rejected machine parts = Proportion * Total number of machine parts
Number of rejected machine parts = 0.0228 * 15,000
Number of rejected machine parts ≈ 342.857
Since we cannot have a fraction of a machine part being rejected, we round up the number of rejected machine parts to the nearest whole number.
So, the correct answer is:
D) 970